Ruzsa's conjecture on divisors in shrinking intervals

Let $n$ be a positive integer and let $\epsilon>0$. Consider the divisors of $n$ in the interval

$$n^{1/2}-n^{1/2-\epsilon}\leq d\leq n^{1/2}+n^{1/2-\epsilon}.$$

Ruzsa's conjecture. There is a constant $K_\epsilon$ such that, for every positive integer $n$, the number of such divisors is at most $K_\epsilon$.

This is described in the source as a stronger question than the Erdős–Rosenfeld conjecture. The source mentions only partial progress for perfect squares and does not state that the general conjecture is resolved.