The least integer not dividing $\binom{2n}{n}$: pin down its typical growth rate (Erdős #731)

For $n\geq 1$ let $m(n)$ be the least integer $m\geq 2$ such that $m\nmid\binom{2n}{n}$. Find some reasonable (explicit) function $f(n)$ such that, for almost all integers $n$ (i.e. on a set of density $1$), $$m(n)\sim f(n).$$

Acceptance. FULLY RESOLVES: an explicit function $f(n)$ together with a proof that $m(n)/f(n)\to 1$ on a set of $n$ of density $1$; a proof determining the almost-sure asymptotic at any equivalent precision (e.g. $m(n)=\exp((c+o(1))(\log n)^{1/2})$ for an explicit constant $c$, if that is the truth) also fully resolves, as does a proof that no such concentration holds (i.e. that $m(n)$ does not have an almost-sure asymptotic). Machine-checkable proof (Lean/Coq) preferred; otherwise a complete written proof with all steps. ADVANCES: (a) any strict sharpening, with proof, of the known window $m(n)=\exp((\log n)^{1/2+o(1)})$ stated in the background (e.g. bounding the second-order term); (b) a large-scale reproducible computation of $m(n)$ (carry-criterion code, stated range $n\leq N$) that extends OEIS A006197 beyond its published terms and delivers an empirically supported explicit conjecture for $f(n)$ with error analysis; (c) a proof of the almost-sure asymptotic restricted to prime-power scales (e.g. determining the typical size of the least prime power failing to divide $\binom{2n}{n}$ for primes in a fixed range). Deliver the proof file, or the code plus data tables and the fitted conjecture.