Anavi–Pollack–Pomerance conjecture on sporadic divisor-sum congruence solutions

Let $k$ be an integer and consider the congruence

$$\sigma(n)\equiv k\pmod n,$$

where $\sigma(n)$ is the sum of the positive divisors of $n$. A solution is regular if $n=pm$, where $p$ is prime, $p\nmid m$, $m\mid\sigma(m)$, and $\sigma(m)=k$; all other solutions are called sporadic. Let $x\geq 3$.

Anavi–Pollack–Pomerance conjecture. Uniformly for integers $k$ satisfying $|k|\leq x/2$, the number of sporadic solutions $n\leq x$ is at most $(\log x)^{O(1)}$.

This conjecture would substantially improve the known bounds for sporadic solutions, which are currently of power-saving type in ranges of $k$. The source notes that the conjecture is motivated by a heuristic concerning the average number of sporadic solutions.