Bound the number of divisors of $n$ in $(\sqrt n,\sqrt n+n^{1/2-\epsilon})$: is it $O_\epsilon(1)$? (Erdős #886)

Let $\epsilon>0$. Is it true that, for all sufficiently large $n$, the number of divisors of $n$ lying in the interval $(n^{1/2},\,n^{1/2}+n^{1/2-\epsilon})$ is $O_\epsilon(1)$? That is, is there a constant $K=K(\epsilon)$, depending only on $\epsilon$ and not on $n$, such that for every large $n$ the count $\#\{d : d\mid n,\ n^{1/2}<d<n^{1/2}+n^{1/2-\epsilon}\}$ is at most $K$? The window here has width $n^{1/2-\epsilon}$, which for small $\epsilon$ is much wider than the $n^{1/4}$ scale on which divisors of $n$ near $\sqrt n$ are understood.

Acceptance. FULLY RESOLVES: a complete proof (machine-checkable in Lean/Coq preferred, otherwise a full written proof with all steps) that for every fixed $\epsilon>0$ there is a constant $K(\epsilon)$ bounding the number of divisors of every large $n$ in $(n^{1/2},\,n^{1/2}+n^{1/2-\epsilon})$; OR a disproof, i.e. a proof that for some $\epsilon>0$ the number of such divisors is unbounded as $n\to\infty$ (which may be supported by, but is not established by, a computational family). ADVANCES: prove a bound for the wide window that is uniform in a nontrivial range of window widths beyond the $n^{1/4}$ scale to which the current $1+C^2$ result of [ErRo97] applies (strictly wider than the best window stated in the background), with proof; OR exhibit, via a reproducible search with an exhaustiveness certificate, integers $n$ carrying an unusually large number of divisors in a window of width $n^{1/2-\epsilon}$, quantifying how the maximal count grows and thereby lower-bounding any admissible $K(\epsilon)$. Deliver the proof file, or the improved-window proof, or the search code plus the tabulated extremal divisor counts and certificate.