YES

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


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

After renaming modulo { (0,0)->0, (0,1)->1, (1,1)->2, (1,0)->3, (0,3)->4, (1,3)->5, (2,0)->6, (2,1)->7, (2,3)->8 },
it remains to prove termination of the 27-rule system
{ 0 0 0 -> 1 2 2 2 3 ,
  0 0 1 -> 1 2 2 2 2 ,
  0 0 4 -> 1 2 2 2 5 ,
  3 0 0 -> 2 2 2 2 3 ,
  3 0 1 -> 2 2 2 2 2 ,
  3 0 4 -> 2 2 2 2 5 ,
  6 0 0 -> 7 2 2 2 3 ,
  6 0 1 -> 7 2 2 2 2 ,
  6 0 4 -> 7 2 2 2 5 ,
  1 3 -> 0 ,
  1 2 -> 1 ,
  1 5 -> 4 ,
  2 3 -> 3 ,
  2 2 -> 2 ,
  2 5 -> 5 ,
  7 3 -> 6 ,
  7 2 -> 7 ,
  7 5 -> 8 ,
  0 1 3 0 -> 0 0 0 0 ,
  0 1 3 1 -> 0 0 0 1 ,
  0 1 3 4 -> 0 0 0 4 ,
  3 1 3 0 -> 3 0 0 0 ,
  3 1 3 1 -> 3 0 0 1 ,
  3 1 3 4 -> 3 0 0 4 ,
  6 1 3 0 -> 6 0 0 0 ,
  6 1 3 1 -> 6 0 0 1 ,
  6 1 3 4 -> 6 0 0 4 }


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 5 |
| 0 1 |
 \   /
1 is interpreted by
 /   \
| 1 5 |
| 0 1 |
 \   /
2 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
3 is interpreted by
 /   \
| 1 6 |
| 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 0 |
| 0 1 |
 \   /
8 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.