YES

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


Applying sparse 2-untiling [Hofbauer/Geser/Waldmann, FSCD 2019].

After renaming modulo { 0->0, 1->1, 2->2, 3->3 },
it remains to prove termination of the 6-rule system
{ 0 0 -> 1 1 1 ,
  0 -> 2 3 ,
  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.