Problem #852: Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime

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