YES

After renaming modulo { a->0, b->1, c->2, d->3 }, 
it remains to prove termination of the 5-rule system
{ 0 0 -> 1 1 1 ,
  1 1 -> 2 2 2 ,
  2 2 -> 3 3 3 ,
  1 -> 3 3 ,
  2 3 3 -> 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 80 |
|  0  1 |
 \     /
1 is interpreted by
 /     \
|  1 53 |
|  0  1 |
 \     /
2 is interpreted by
 /     \
|  1 35 |
|  0  1 |
 \     /
3 is interpreted by
 /     \
|  1 23 |
|  0  1 |
 \     /

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

The system is trivially terminating.