YES

After renaming modulo { 1->0, 2->1, 0->2, L->3, R->4 }, 
it remains to prove termination of the 8-rule system
{ 0 1 0 -> 1 2 1 ,
  2 1 0 -> 0 2 1 ,
  3 1 0 -> 3 0 2 1 ,
  0 1 2 -> 1 2 0 ,
  0 1 4 -> 1 2 0 4 ,
  2 1 2 -> 0 2 0 ,
  3 1 2 -> 3 0 2 0 ,
  2 1 4 -> 0 2 0 4 }


The system was reversed.

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


Applying the dependency pairs transformation.

After renaming modulo { (0,true)->0, (1,false)->1, (0,false)->2, (2,true)->3, (2,false)->4, (3,false)->5, (4,true)->6, (4,false)->7 },
it remains to prove termination of the 28-rule system
{ 0 1 2 -> 3 1 ,
  0 1 4 -> 3 2 ,
  0 1 4 -> 0 ,
  0 1 5 -> 3 2 5 ,
  0 1 5 -> 0 5 ,
  3 1 2 -> 0 4 1 ,
  3 1 2 -> 3 1 ,
  6 1 2 -> 6 2 4 1 ,
  6 1 2 -> 0 4 1 ,
  6 1 2 -> 3 1 ,
  3 1 4 -> 0 4 2 ,
  3 1 4 -> 3 2 ,
  3 1 4 -> 0 ,
  3 1 5 -> 0 4 2 5 ,
  3 1 5 -> 3 2 5 ,
  3 1 5 -> 0 5 ,
  6 1 4 -> 6 2 4 2 ,
  6 1 4 -> 0 4 2 ,
  6 1 4 -> 3 2 ,
  6 1 4 -> 0 ,
  2 1 2 ->= 1 4 1 ,
  2 1 4 ->= 1 4 2 ,
  2 1 5 ->= 1 4 2 5 ,
  4 1 2 ->= 2 4 1 ,
  7 1 2 ->= 7 2 4 1 ,
  4 1 4 ->= 2 4 2 ,
  4 1 5 ->= 2 4 2 5 ,
  7 1 4 ->= 7 2 4 2 }


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 1 |
| 0 1 |
 \   /
7 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 }, 
it remains to prove termination of the 23-rule system
{ 0 1 2 -> 3 1 ,
  0 1 4 -> 3 2 ,
  0 1 4 -> 0 ,
  0 1 5 -> 3 2 5 ,
  0 1 5 -> 0 5 ,
  3 1 2 -> 0 4 1 ,
  3 1 2 -> 3 1 ,
  6 1 2 -> 6 2 4 1 ,
  3 1 4 -> 0 4 2 ,
  3 1 4 -> 3 2 ,
  3 1 4 -> 0 ,
  3 1 5 -> 0 4 2 5 ,
  3 1 5 -> 3 2 5 ,
  3 1 5 -> 0 5 ,
  6 1 4 -> 6 2 4 2 ,
  2 1 2 ->= 1 4 1 ,
  2 1 4 ->= 1 4 2 ,
  2 1 5 ->= 1 4 2 5 ,
  4 1 2 ->= 2 4 1 ,
  7 1 2 ->= 7 2 4 1 ,
  4 1 4 ->= 2 4 2 ,
  4 1 5 ->= 2 4 2 5 ,
  7 1 4 ->= 7 2 4 2 }


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

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


The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 6:

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

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

The system is trivially terminating.