YES

After renaming modulo { 1->0, 4->1, 3->2, 2->3, 5->4, 6->5 }, 
it remains to prove termination of the 9-rule system
{ 0 0 -> 1 2 ,
  0 3 -> 3 0 ,
  3 3 -> 0 0 0 ,
  2 2 -> 4 5 ,
  2 1 -> 0 0 ,
  1 1 -> 2 ,
  4 4 -> 5 3 ,
  4 5 -> 0 3 ,
  5 5 -> 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 85 |
|  0  1 |
 \     /
1 is interpreted by
 /     \
|  1 57 |
|  0  1 |
 \     /
2 is interpreted by
 /       \
|   1 113 |
|   0   1 |
 \       /
3 is interpreted by
 /       \
|   1 128 |
|   0   1 |
 \       /
4 is interpreted by
 /       \
|   1 118 |
|   0   1 |
 \       /
5 is interpreted by
 /       \
|   1 107 |
|   0   1 |
 \       /

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


Applying the dependency pairs transformation.

After renaming modulo { (0,true)->0, (0,false)->1, (2,true)->2, (3,false)->3, (1,false)->4, (2,false)->5 },
it remains to prove termination of the 7-rule system
{ 0 1 -> 2 ,
  0 3 -> 0 ,
  2 4 -> 0 1 ,
  2 4 -> 0 ,
  1 1 ->= 4 5 ,
  1 3 ->= 3 1 ,
  5 4 ->= 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 1 |
| 0 1 |
 \   /
5 is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /

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


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

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

The system is trivially terminating.