YES

After renaming modulo { a->0, g->1, d->2, b->3, c->4 }, 
it remains to prove termination of the 5-rule system
{ 0 -> 1 2 ,
  3 3 3 -> 4 2 4 ,
  3 3 -> 0 1 1 ,
  4 2 -> 1 1 ,
  1 1 1 -> 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 6 |
| 0 1 |
 \   /
1 is interpreted by
 /   \
| 1 6 |
| 0 1 |
 \   /
2 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
3 is interpreted by
 /   \
| 1 9 |
| 0 1 |
 \   /
4 is interpreted by
 /     \
|  1 13 |
|  0  1 |
 \     /

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


The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 4:

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

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


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 1 |
| 0 1 |
 \   /

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

The system is trivially terminating.