YES

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


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 30 |
|  0  1 |
 \     /
1 is interpreted by
 /     \
|  1 27 |
|  0  1 |
 \     /
2 is interpreted by
 /     \
|  1 32 |
|  0  1 |
 \     /
3 is interpreted by
 /     \
|  1 21 |
|  0  1 |
 \     /

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

The system is trivially terminating.