YES

After renaming modulo { 0->0, *->1, 1->2, #->3, $->4 }, 
it remains to prove termination of the 7-rule system
{ 0 0 1 1 -> 1 1 2 2 ,
  2 2 1 1 -> 0 0 3 3 ,
  3 3 0 0 -> 0 0 3 3 ,
  3 3 2 2 -> 2 2 3 3 ,
  3 3 4 4 -> 1 1 4 4 ,
  3 3 3 3 -> 3 3 ,
  3 3 1 1 -> 1 1 }


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 1 |
| 0 1 |
 \   /
1 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
2 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
3 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
4 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /

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


Applying the dependency pairs transformation.

After renaming modulo { (0,true)->0, (0,false)->1, (1,false)->2, (2,true)->3, (2,false)->4, (3,false)->5, (3,true)->6, (4,false)->7 },
it remains to prove termination of the 19-rule system
{ 0 1 2 2 -> 3 4 ,
  0 1 2 2 -> 3 ,
  3 4 2 2 -> 0 1 5 5 ,
  3 4 2 2 -> 0 5 5 ,
  3 4 2 2 -> 6 5 ,
  3 4 2 2 -> 6 ,
  6 5 1 1 -> 0 1 5 5 ,
  6 5 1 1 -> 0 5 5 ,
  6 5 1 1 -> 6 5 ,
  6 5 1 1 -> 6 ,
  6 5 4 4 -> 3 4 5 5 ,
  6 5 4 4 -> 3 5 5 ,
  6 5 4 4 -> 6 5 ,
  6 5 4 4 -> 6 ,
  1 1 2 2 ->= 2 2 4 4 ,
  4 4 2 2 ->= 1 1 5 5 ,
  5 5 1 1 ->= 1 1 5 5 ,
  5 5 4 4 ->= 4 4 5 5 ,
  5 5 7 7 ->= 2 2 7 7 }


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 1 |
| 0 1 |
 \   /
1 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
2 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
3 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
4 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
5 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
6 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
7 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /

After renaming modulo { 3->0, 4->1, 2->2, 0->3, 1->4, 5->5, 6->6, 7->7 }, 
it remains to prove termination of the 8-rule system
{ 0 1 2 2 -> 3 4 5 5 ,
  6 5 4 4 -> 3 4 5 5 ,
  6 5 1 1 -> 0 1 5 5 ,
  4 4 2 2 ->= 2 2 1 1 ,
  1 1 2 2 ->= 4 4 5 5 ,
  5 5 4 4 ->= 4 4 5 5 ,
  5 5 1 1 ->= 1 1 5 5 ,
  5 5 7 7 ->= 2 2 7 7 }


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 1 |
| 0 1 |
 \   /
1 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
2 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
3 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
4 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
5 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
6 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
7 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /

After renaming modulo { 6->0, 5->1, 1->2, 0->3, 4->4, 2->5, 7->6 }, 
it remains to prove termination of the 6-rule system
{ 0 1 2 2 -> 3 2 1 1 ,
  4 4 5 5 ->= 5 5 2 2 ,
  2 2 5 5 ->= 4 4 1 1 ,
  1 1 4 4 ->= 4 4 1 1 ,
  1 1 2 2 ->= 2 2 1 1 ,
  1 1 6 6 ->= 5 5 6 6 }


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 1 |
| 0 1 |
 \   /
1 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
2 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
3 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
4 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
5 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
6 is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /

After renaming modulo { 4->0, 5->1, 2->2, 1->3, 6->4 }, 
it remains to prove termination of the 5-rule system
{ 0 0 1 1 ->= 1 1 2 2 ,
  2 2 1 1 ->= 0 0 3 3 ,
  3 3 0 0 ->= 0 0 3 3 ,
  3 3 2 2 ->= 2 2 3 3 ,
  3 3 4 4 ->= 1 1 4 4 }

The system is trivially terminating.