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With arbitrary generation rules they are surely not. This is a counter example on 4x4:

  | 1 | 1 | 1 | 1 |
  | 1 | 3 | 3 | 3 |
  | 1 | 3 | 2 | 2 | 
  | 1 | 3 | 2 | G |
Or

  | 2 | 2 | 2 | 2 |
  | 2 | 2 | 2 | 1 |
  | 2 | 2 | 2 | 2 | 
  | 2 | 1 | 2 | G |
This seems to be able to be understood as a reachability graph problem of some sort perhaps.

Edit: formatting






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