Runs of $>c_1\log x$ consecutive primes with all gaps $>c_2$: must they always exist? (Erdős #238)

Let $c_1,c_2>0$. Is it true that for every sufficiently large $x$ there exist more than $c_1\log x$ consecutive primes, all at most $x$, such that the difference between any two of them exceeds $c_2$? Equivalently: does there always exist a block of more than $c_1\log x$ consecutive primes $p_m,p_{m+1},\ldots\le x$ in which every gap $p_{i+1}-p_i>c_2$? The assertion is to hold for all pairs $c_1,c_2>0$.

Acceptance. FULLY RESOLVES: a complete proof that for every $c_1,c_2>0$ and all sufficiently large $x$ there exist more than $c_1\log x$ consecutive primes $\le x$ with all pairwise differences $>c_2$ — machine-checkable (Lean/Coq) preferred, otherwise a full written proof; OR a disproof exhibiting $c_1,c_2>0$ for which the property fails for arbitrarily large $x$. ADVANCES (each a complete proof, strictly beyond Erdős's small-$c_1$ result stated in the background): prove the conclusion for all $c_1$ below an explicit threshold larger than any previously admissible value, with the dependence on $c_2$ made explicit; or establish it for all $c_1>0$ under a clearly stated hypothesis (e.g. a form of Hardy–Littlewood); or prove a quantitative lower bound on the length of the longest such run in $[1,x]$ that grows strictly faster than the currently guaranteed $c_1\log x$. Deliver a machine-checkable or complete written proof.