The EGPS conjecture on preimages of density-zero sets under the proper-divisor sum

Let $s(n)=\sum_{d\mid n,\ d<n}d$ be the sum-of-proper-divisors function, and let $\mathcal{A}$ be a set of integers with asymptotic density zero, meaning that

$$\lim_{x\to\infty}\frac{1}{x}\#\{n\leq x:n\in\mathcal{A}\}=0.$$

EGPS conjecture. The preimage $s^{-1}(\mathcal{A})$ also has asymptotic density zero.

The conjecture concerns whether the proper-divisor sum can produce a positive-density set of inputs from a density-zero target set. The full EGPS conjecture is still open today.