YES

After renaming modulo { A->0, b->1, a->2, B->3 }, 
it remains to prove termination of the 4-rule system
{ 0 1 -> 1 2 3 0 ,
  3 2 -> 2 1 0 3 ,
  0 2 -> ,
  3 1 -> }


Applying sparse 2-tiling [Hofbauer/Geser/Waldmann, FSCD 2019].

After renaming modulo { (0,0)->0, (0,1)->1, (1,0)->2, (1,2)->3, (2,3)->4, (3,0)->5, (1,1)->6, (0,2)->7, (1,3)->8, (0,3)->9, (1,5)->10, (0,5)->11, (2,0)->12, (2,1)->13, (3,1)->14, (3,2)->15, (2,2)->16, (3,3)->17, (2,5)->18, (3,5)->19 },
it remains to prove termination of the 80-rule system
{ 0 1 2 -> 1 3 4 5 0 ,
  0 1 6 -> 1 3 4 5 1 ,
  0 1 3 -> 1 3 4 5 7 ,
  0 1 8 -> 1 3 4 5 9 ,
  0 1 10 -> 1 3 4 5 11 ,
  2 1 2 -> 6 3 4 5 0 ,
  2 1 6 -> 6 3 4 5 1 ,
  2 1 3 -> 6 3 4 5 7 ,
  2 1 8 -> 6 3 4 5 9 ,
  2 1 10 -> 6 3 4 5 11 ,
  12 1 2 -> 13 3 4 5 0 ,
  12 1 6 -> 13 3 4 5 1 ,
  12 1 3 -> 13 3 4 5 7 ,
  12 1 8 -> 13 3 4 5 9 ,
  12 1 10 -> 13 3 4 5 11 ,
  5 1 2 -> 14 3 4 5 0 ,
  5 1 6 -> 14 3 4 5 1 ,
  5 1 3 -> 14 3 4 5 7 ,
  5 1 8 -> 14 3 4 5 9 ,
  5 1 10 -> 14 3 4 5 11 ,
  9 15 12 -> 7 13 2 9 5 ,
  9 15 13 -> 7 13 2 9 14 ,
  9 15 16 -> 7 13 2 9 15 ,
  9 15 4 -> 7 13 2 9 17 ,
  9 15 18 -> 7 13 2 9 19 ,
  8 15 12 -> 3 13 2 9 5 ,
  8 15 13 -> 3 13 2 9 14 ,
  8 15 16 -> 3 13 2 9 15 ,
  8 15 4 -> 3 13 2 9 17 ,
  8 15 18 -> 3 13 2 9 19 ,
  4 15 12 -> 16 13 2 9 5 ,
  4 15 13 -> 16 13 2 9 14 ,
  4 15 16 -> 16 13 2 9 15 ,
  4 15 4 -> 16 13 2 9 17 ,
  4 15 18 -> 16 13 2 9 19 ,
  17 15 12 -> 15 13 2 9 5 ,
  17 15 13 -> 15 13 2 9 14 ,
  17 15 16 -> 15 13 2 9 15 ,
  17 15 4 -> 15 13 2 9 17 ,
  17 15 18 -> 15 13 2 9 19 ,
  0 7 12 -> 0 ,
  0 7 13 -> 1 ,
  0 7 16 -> 7 ,
  0 7 4 -> 9 ,
  0 7 18 -> 11 ,
  2 7 12 -> 2 ,
  2 7 13 -> 6 ,
  2 7 16 -> 3 ,
  2 7 4 -> 8 ,
  2 7 18 -> 10 ,
  12 7 12 -> 12 ,
  12 7 13 -> 13 ,
  12 7 16 -> 16 ,
  12 7 4 -> 4 ,
  12 7 18 -> 18 ,
  5 7 12 -> 5 ,
  5 7 13 -> 14 ,
  5 7 16 -> 15 ,
  5 7 4 -> 17 ,
  5 7 18 -> 19 ,
  9 14 2 -> 0 ,
  9 14 6 -> 1 ,
  9 14 3 -> 7 ,
  9 14 8 -> 9 ,
  9 14 10 -> 11 ,
  8 14 2 -> 2 ,
  8 14 6 -> 6 ,
  8 14 3 -> 3 ,
  8 14 8 -> 8 ,
  8 14 10 -> 10 ,
  4 14 2 -> 12 ,
  4 14 6 -> 13 ,
  4 14 3 -> 16 ,
  4 14 8 -> 4 ,
  4 14 10 -> 18 ,
  17 14 2 -> 5 ,
  17 14 6 -> 14 ,
  17 14 3 -> 15 ,
  17 14 8 -> 17 ,
  17 14 10 -> 19 }


The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 2:

0 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
1 is interpreted by
 /   \
| 1 2 |
| 0 1 |
 \   /
2 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
3 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
4 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
5 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
6 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
7 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
8 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
9 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
10 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
11 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
12 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
13 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
14 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
15 is interpreted by
 /   \
| 1 2 |
| 0 1 |
 \   /
16 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
17 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
18 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
19 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /

After renaming modulo { 0->0, 1->1, 2->2, 3->3, 4->4, 5->5, 6->6, 7->7, 14->8, 9->9, 15->10, 13->11, 16->12, 17->13 }, 
it remains to prove termination of the 28-rule system
{ 0 1 2 -> 1 3 4 5 0 ,
  0 1 6 -> 1 3 4 5 1 ,
  0 1 3 -> 1 3 4 5 7 ,
  2 1 2 -> 6 3 4 5 0 ,
  2 1 6 -> 6 3 4 5 1 ,
  2 1 3 -> 6 3 4 5 7 ,
  5 1 2 -> 8 3 4 5 0 ,
  5 1 6 -> 8 3 4 5 1 ,
  5 1 3 -> 8 3 4 5 7 ,
  9 10 11 -> 7 11 2 9 8 ,
  9 10 12 -> 7 11 2 9 10 ,
  9 10 4 -> 7 11 2 9 13 ,
  4 10 11 -> 12 11 2 9 8 ,
  4 10 12 -> 12 11 2 9 10 ,
  4 10 4 -> 12 11 2 9 13 ,
  13 10 11 -> 10 11 2 9 8 ,
  13 10 12 -> 10 11 2 9 10 ,
  13 10 4 -> 10 11 2 9 13 ,
  0 7 11 -> 1 ,
  2 7 11 -> 6 ,
  5 7 11 -> 8 ,
  5 7 12 -> 10 ,
  5 7 4 -> 13 ,
  9 8 2 -> 0 ,
  9 8 6 -> 1 ,
  9 8 3 -> 7 ,
  4 8 3 -> 12 ,
  13 8 3 -> 10 }


Applying the dependency pairs transformation.

After renaming modulo { (0,true)->0, (1,false)->1, (2,false)->2, (4,true)->3, (5,false)->4, (0,false)->5, (5,true)->6, (6,false)->7, (3,false)->8, (7,false)->9, (2,true)->10, (9,true)->11, (10,false)->12, (11,false)->13, (9,false)->14, (8,false)->15, (12,false)->16, (4,false)->17, (13,false)->18, (13,true)->19 },
it remains to prove termination of the 72-rule system
{ 0 1 2 -> 3 4 5 ,
  0 1 2 -> 6 5 ,
  0 1 2 -> 0 ,
  0 1 7 -> 3 4 1 ,
  0 1 7 -> 6 1 ,
  0 1 8 -> 3 4 9 ,
  0 1 8 -> 6 9 ,
  10 1 2 -> 3 4 5 ,
  10 1 2 -> 6 5 ,
  10 1 2 -> 0 ,
  10 1 7 -> 3 4 1 ,
  10 1 7 -> 6 1 ,
  10 1 8 -> 3 4 9 ,
  10 1 8 -> 6 9 ,
  6 1 2 -> 3 4 5 ,
  6 1 2 -> 6 5 ,
  6 1 2 -> 0 ,
  6 1 7 -> 3 4 1 ,
  6 1 7 -> 6 1 ,
  6 1 8 -> 3 4 9 ,
  6 1 8 -> 6 9 ,
  11 12 13 -> 10 14 15 ,
  11 12 13 -> 11 15 ,
  11 12 16 -> 10 14 12 ,
  11 12 16 -> 11 12 ,
  11 12 17 -> 10 14 18 ,
  11 12 17 -> 11 18 ,
  11 12 17 -> 19 ,
  3 12 13 -> 10 14 15 ,
  3 12 13 -> 11 15 ,
  3 12 16 -> 10 14 12 ,
  3 12 16 -> 11 12 ,
  3 12 17 -> 10 14 18 ,
  3 12 17 -> 11 18 ,
  3 12 17 -> 19 ,
  19 12 13 -> 10 14 15 ,
  19 12 13 -> 11 15 ,
  19 12 16 -> 10 14 12 ,
  19 12 16 -> 11 12 ,
  19 12 17 -> 10 14 18 ,
  19 12 17 -> 11 18 ,
  19 12 17 -> 19 ,
  6 9 17 -> 19 ,
  11 15 2 -> 0 ,
  5 1 2 ->= 1 8 17 4 5 ,
  5 1 7 ->= 1 8 17 4 1 ,
  5 1 8 ->= 1 8 17 4 9 ,
  2 1 2 ->= 7 8 17 4 5 ,
  2 1 7 ->= 7 8 17 4 1 ,
  2 1 8 ->= 7 8 17 4 9 ,
  4 1 2 ->= 15 8 17 4 5 ,
  4 1 7 ->= 15 8 17 4 1 ,
  4 1 8 ->= 15 8 17 4 9 ,
  14 12 13 ->= 9 13 2 14 15 ,
  14 12 16 ->= 9 13 2 14 12 ,
  14 12 17 ->= 9 13 2 14 18 ,
  17 12 13 ->= 16 13 2 14 15 ,
  17 12 16 ->= 16 13 2 14 12 ,
  17 12 17 ->= 16 13 2 14 18 ,
  18 12 13 ->= 12 13 2 14 15 ,
  18 12 16 ->= 12 13 2 14 12 ,
  18 12 17 ->= 12 13 2 14 18 ,
  5 9 13 ->= 1 ,
  2 9 13 ->= 7 ,
  4 9 13 ->= 15 ,
  4 9 16 ->= 12 ,
  4 9 17 ->= 18 ,
  14 15 2 ->= 5 ,
  14 15 7 ->= 1 ,
  14 15 8 ->= 9 ,
  17 15 8 ->= 16 ,
  18 15 8 ->= 12 }


The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 2:

0 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
1 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
2 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
3 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
4 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
5 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
6 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
7 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
8 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
9 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
10 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
11 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
12 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
13 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
14 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
15 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
16 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
17 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
18 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
19 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /

After renaming modulo { 5->0, 1->1, 2->2, 8->3, 17->4, 4->5, 7->6, 9->7, 15->8, 14->9, 12->10, 13->11, 16->12, 18->13 }, 
it remains to prove termination of the 28-rule system
{ 0 1 2 ->= 1 3 4 5 0 ,
  0 1 6 ->= 1 3 4 5 1 ,
  0 1 3 ->= 1 3 4 5 7 ,
  2 1 2 ->= 6 3 4 5 0 ,
  2 1 6 ->= 6 3 4 5 1 ,
  2 1 3 ->= 6 3 4 5 7 ,
  5 1 2 ->= 8 3 4 5 0 ,
  5 1 6 ->= 8 3 4 5 1 ,
  5 1 3 ->= 8 3 4 5 7 ,
  9 10 11 ->= 7 11 2 9 8 ,
  9 10 12 ->= 7 11 2 9 10 ,
  9 10 4 ->= 7 11 2 9 13 ,
  4 10 11 ->= 12 11 2 9 8 ,
  4 10 12 ->= 12 11 2 9 10 ,
  4 10 4 ->= 12 11 2 9 13 ,
  13 10 11 ->= 10 11 2 9 8 ,
  13 10 12 ->= 10 11 2 9 10 ,
  13 10 4 ->= 10 11 2 9 13 ,
  0 7 11 ->= 1 ,
  2 7 11 ->= 6 ,
  5 7 11 ->= 8 ,
  5 7 12 ->= 10 ,
  5 7 4 ->= 13 ,
  9 8 2 ->= 0 ,
  9 8 6 ->= 1 ,
  9 8 3 ->= 7 ,
  4 8 3 ->= 12 ,
  13 8 3 ->= 10 }

The system is trivially terminating.