Estimate h(n): shortest interval holding distinct multiples of each of the first π(n) primes (Erdős #860)

Let $\pi(n)$ denote the number of primes not exceeding $n$, and let $p_i$ denote the $i$th prime. Define $h(n)$ to be the least value such that, for every integer $m\geq 1$, the interval $(m,m+h(n))$ contains distinct integers $a_1,\ldots,a_{\pi(n)}$ with $p_i\mid a_i$ for each $1\leq i\leq \pi(n)$ — that is, a system of distinct representatives in which the $i$th prime divides the $i$th chosen integer. Estimate $h(n)$; in particular, determine its order of growth.

Acceptance. This is an OPEN, proof-shaped estimation problem. FULLY RESOLVES: determine the true order of growth of $h(n)$ — a complete proof of matching upper and lower bounds, i.e. $h(n)\asymp f(n)$ (or an asymptotic $h(n)\sim f(n)$) for an explicit $f$, thereby pinning the rate between the current lower bound (Erdős–Selfridge $(3-o(1))n$, strengthened by Ruzsa to $h(n)/n\to\infty$) and the upper bound $n^{3/2}/(\log n)^{1/2}$. ADVANCES: strictly improve either frontier bound stated in the background — a proof of an upper bound smaller than $n^{3/2}/(\log n)^{1/2}$, or of a lower bound larger than $(3-o(1))n$ (for instance $h(n)\gg n\psi(n)$ with an explicit $\psi\to\infty$ quantifying Ruzsa's result), each with proof; or rigorously extend the table of exact values of $h(n)$ for small $n$ with a reproducible matching-based certificate of exhaustiveness. Deliver the proof(s), or the verified-values program together with its certificates.