YES

After renaming modulo { b->0, c->1, d->2, a->3 }, 
it remains to prove termination of the 5-rule system
{ 0 0 -> 1 2 ,
  1 1 -> 2 2 2 ,
  2 2 2 -> 3 1 ,
  3 3 ->= 0 1 ,
  0 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 22 |
|  0  1 |
 \     /
1 is interpreted by
 /     \
|  1 26 |
|  0  1 |
 \     /
2 is interpreted by
 /     \
|  1 17 |
|  0  1 |
 \     /
3 is interpreted by
 /     \
|  1 24 |
|  0  1 |
 \     /

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

The system is trivially terminating.