Longest run of distinct consecutive prime gaps: estimate $h(x)$ (Erdős #852)

Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Let $h(x)$ be the largest integer $H$ such that for some $n<x$ the consecutive gaps $d_n,d_{n+1},\ldots,d_{n+H-1}$ are all distinct. Estimate $h(x)$. In particular, is it true that $$h(x)>(\log x)^c$$ for some constant $c>0$, and that $$h(x)=o(\log x)?$$

Acceptance. FULLY RESOLVES: a rigorous determination of the order of $h(x)$ — in particular a proof of both $h(x)>(\log x)^c$ for some explicit $c>0$ and $h(x)=o(\log x)$, or a proof refuting either — with complete justification. ADVANCES: prove a power lower bound $h(x)>(\log x)^c$ for an explicit $c>0$, strengthening the qualitative $h(x)\to\infty$ from Brun's sieve stated in the background; or prove any nontrivial upper bound such as $h(x)=o((\log x)^A)$ for a stated $A$, or $h(x)=o(\log x)$; or computationally extend the record longest run of distinct consecutive prime gaps to a new bound, supplying the scanning program, the witnessing starting index $n$, the value of $h(x)$ at a stated $x$, and the extended OEIS data. Deliver the proof or the computation code with the extended records/sequence.