YES

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


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