YES

After renaming modulo { a->0, b->1, c->2, d->3, f->4, g->5 }, 
it remains to prove termination of the 8-rule system
{ 0 0 -> 1 2 ,
  1 1 -> 2 3 ,
  1 -> 0 ,
  2 2 -> 3 4 ,
  3 3 -> 4 4 4 ,
  3 -> 1 ,
  4 4 -> 5 0 ,
  5 5 -> 0 }


The system was reversed.

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


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

0 is interpreted by
 /       \
|   1 155 |
|   0   1 |
 \       /
1 is interpreted by
 /       \
|   1 147 |
|   0   1 |
 \       /
2 is interpreted by
 /       \
|   1 162 |
|   0   1 |
 \       /
3 is interpreted by
 /       \
|   1 176 |
|   0   1 |
 \       /
4 is interpreted by
 /       \
|   1 117 |
|   0   1 |
 \       /
5 is interpreted by
 /     \
|  1 78 |
|  0  1 |
 \     /

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

The system is trivially terminating.