Conjecture on polylogarithmically many solutions to σ(n)=kn+a

Let $\sigma(n)$ denote the sum-of-divisors function. For integers $k\geq 2$, $x\geq 3$, and $a\in\mathbb{Z}$ satisfying

$$|a|\leq \frac{x}{2},$$

consider the solutions $n\leq x$ of $\sigma(n)=kn+a$. Polylogarithmic solution-count conjecture. The number of such solutions is

$$\ll (\log x)^C,$$

where the implied constant and $C$ are absolute constants. The conjecture refines expectations about the number of solutions to generalized divisor-sum equations after excluding the problematic cases $k=0$ and $k=1$; the source later shows that this bound fails, so the conjecture is refuted.