YES

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


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

After renaming modulo { (0,0)->0, (0,1)->1, (1,0)->2, (0,3)->3, (1,1)->4, (1,3)->5, (2,1)->6 },
it remains to prove termination of the 24-rule system
{ 0 0 0 0 -> 1 2 0 ,
  0 0 0 1 -> 1 2 1 ,
  0 0 0 3 -> 1 2 3 ,
  2 0 0 0 -> 4 2 0 ,
  2 0 0 1 -> 4 2 1 ,
  2 0 0 3 -> 4 2 3 ,
  1 4 2 -> 1 2 1 2 ,
  1 4 4 -> 1 2 1 4 ,
  1 4 5 -> 1 2 1 5 ,
  4 4 2 -> 4 2 1 2 ,
  4 4 4 -> 4 2 1 4 ,
  4 4 5 -> 4 2 1 5 ,
  6 4 2 -> 6 2 1 2 ,
  6 4 4 -> 6 2 1 4 ,
  6 4 5 -> 6 2 1 5 ,
  1 4 4 2 -> 1 2 0 0 ,
  1 4 4 4 -> 1 2 0 1 ,
  1 4 4 5 -> 1 2 0 3 ,
  4 4 4 2 -> 4 2 0 0 ,
  4 4 4 4 -> 4 2 0 1 ,
  4 4 4 5 -> 4 2 0 3 ,
  6 4 4 2 -> 6 2 0 0 ,
  6 4 4 4 -> 6 2 0 1 ,
  6 4 4 5 -> 6 2 0 3 }


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 0 |
| 0 1 |
 \   /
2 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
3 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
4 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
5 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
6 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /

After renaming modulo { 1->0, 4->1, 2->2, 0->3, 6->4 }, 
it remains to prove termination of the 3-rule system
{ 0 1 1 2 -> 0 2 3 3 ,
  1 1 1 2 -> 1 2 3 3 ,
  4 1 1 2 -> 4 2 3 3 }


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 0 |
| 0 1 |
 \   /
3 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
4 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /

After renaming modulo {  }, 
it remains to prove termination of the 0-rule system
{ }

The system is trivially terminating.