Finiteness conjecture for power-sum congruence solutions

Let $m\in\mathbb{N}$, and let $S_n(n)=\sum_{i=1}^{n}i^n$. The solutions are the natural numbers $n$ satisfying

$$S_n(n)\equiv m\pmod{n}.$$

Finiteness conjecture. For every $m\in\mathbb{N}$, the set of solutions to the congruence

$$S_n(n)\equiv m\pmod{n}$$

is finite.

The conjecture is proposed as a natural extension of the paper's results for prime $m$; the authors state that new ideas are needed to handle the general composite case, and no resolution is given.