Order of magnitude of Jacobsthal's function $h(k)$: is $h(k)\ll k^2$? (Erdős #970)

For a positive integer $n$, let $g(n)$ be the least $m$ such that every run of $m$ consecutive integers contains one coprime to $n$ (the maximal gap between integers coprime to $n$). Jacobsthal's function is $$h(k)=\max_{\omega(n)\le k} g(n),$$ the maximum of $g(n)$ over all $n$ with at most $k$ distinct prime factors $\omega(n)$ (the maximum is attained at products of the first $k$ primes). Determine the order of magnitude of $h(k)$. In particular, is Jacobsthal's conjecture $h(k)\ll k^2$ true?

Acceptance. FULLY RESOLVES: a proof (machine-checkable preferred, else a complete written proof) determining the order of magnitude of $h(k)$ — in particular a proof of Jacobsthal's conjecture $h(k)\ll k^2$ (matched against the easy lower bound $h(k)\gg k$), or a rigorous disproof exhibiting faster growth. ADVANCES (proof or reproducible certificate required; must strictly beat the bound stated in the background): (a) improve the upper bound below $(k\log k)^2$ of Iwaniec; (b) improve the lower bound beyond the Ford–Green–Konyagin–Maynard–Tao bound stated above; or (c) compute new exact values of $h(k)$ beyond the currently tabulated range, via a reproducible search program plus a certificate that the reported run length is maximal (extending OEIS A048669 / A048670). Deliver the proof/formalization, or the search code plus the new certified values.