YES

After renaming modulo { 1->0, q0->1, 0->2, q1->3, q2->4 }, 
it remains to prove termination of the 7-rule system
{ 0 1 0 -> 2 0 3 ,
  0 1 2 -> 2 2 3 ,
  0 3 0 -> 0 0 3 ,
  0 3 2 -> 0 2 3 ,
  2 3 -> 4 0 ,
  0 4 -> 4 0 ,
  2 4 -> 2 1 }


The system was reversed.

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


Applying the dependency pairs transformation.

After renaming modulo { (0,true)->0, (1,false)->1, (0,false)->2, (2,true)->3, (3,false)->4, (3,true)->5, (2,false)->6, (4,false)->7, (4,true)->8 },
it remains to prove termination of the 24-rule system
{ 0 1 2 -> 3 2 4 ,
  0 1 2 -> 0 4 ,
  0 1 2 -> 5 ,
  5 1 2 -> 3 4 4 ,
  5 1 2 -> 5 4 ,
  5 1 2 -> 5 ,
  0 6 2 -> 3 2 2 ,
  0 6 2 -> 0 2 ,
  0 6 2 -> 0 ,
  5 6 2 -> 3 4 2 ,
  5 6 2 -> 5 2 ,
  5 6 2 -> 0 ,
  3 4 -> 0 7 ,
  3 4 -> 8 ,
  8 2 -> 0 7 ,
  8 2 -> 8 ,
  8 4 -> 5 ,
  2 1 2 ->= 6 2 4 ,
  4 1 2 ->= 6 4 4 ,
  2 6 2 ->= 6 2 2 ,
  4 6 2 ->= 6 4 2 ,
  6 4 ->= 2 7 ,
  7 2 ->= 2 7 ,
  7 4 ->= 1 4 }


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 1 |
| 0 1 |
 \   /
8 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /

After renaming modulo { 0->0, 1->1, 2->2, 3->3, 4->4, 5->5, 6->6, 7->7, 8->8 }, 
it remains to prove termination of the 13-rule system
{ 0 1 2 -> 3 2 4 ,
  5 1 2 -> 3 4 4 ,
  0 6 2 -> 3 2 2 ,
  5 6 2 -> 3 4 2 ,
  3 4 -> 0 7 ,
  8 2 -> 0 7 ,
  2 1 2 ->= 6 2 4 ,
  4 1 2 ->= 6 4 4 ,
  2 6 2 ->= 6 2 2 ,
  4 6 2 ->= 6 4 2 ,
  6 4 ->= 2 7 ,
  7 2 ->= 2 7 ,
  7 4 ->= 1 4 }


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

After renaming modulo { 0->0, 1->1, 2->2, 3->3, 4->4, 6->5, 7->6 }, 
it remains to prove termination of the 10-rule system
{ 0 1 2 -> 3 2 4 ,
  0 5 2 -> 3 2 2 ,
  3 4 -> 0 6 ,
  2 1 2 ->= 5 2 4 ,
  4 1 2 ->= 5 4 4 ,
  2 5 2 ->= 5 2 2 ,
  4 5 2 ->= 5 4 2 ,
  5 4 ->= 2 6 ,
  6 2 ->= 2 6 ,
  6 4 ->= 1 4 }


The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 3:

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

After renaming modulo { 0->0, 1->1, 2->2, 3->3, 4->4, 5->5, 6->6 }, 
it remains to prove termination of the 9-rule system
{ 0 1 2 -> 3 2 4 ,
  0 5 2 -> 3 2 2 ,
  2 1 2 ->= 5 2 4 ,
  4 1 2 ->= 5 4 4 ,
  2 5 2 ->= 5 2 2 ,
  4 5 2 ->= 5 4 2 ,
  5 4 ->= 2 6 ,
  6 2 ->= 2 6 ,
  6 4 ->= 1 4 }


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 |
 \   /

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

The system is trivially terminating.