Erdős–Granville–Pomerance–Spiro conjecture on preimages under the sum-of-proper-divisors function

Let $s(n)$ denote the sum of the proper divisors of $n$, and let $\mathcal{A}$ be a set of integers. The preimage $s^{-1}(\mathcal{A})$ is the set of integers $n$ such that $s(n)\in\mathcal{A}$. A set of integers has asymptotic density zero if its natural density exists and equals zero.

Erdős–Granville–Pomerance–Spiro conjecture. If $\mathcal{A}$ has asymptotic density zero, then $s^{-1}(\mathcal{A})$ also has asymptotic density zero.

This conjecture concerns how the sum-of-proper-divisors function transforms sparse sets. It remains open, although it has been proved in some special cases.