The $k_{\max}$-conjecture for integer roots of Moser polynomials

Let $F_{s,k}(n)$ be the Moser polynomial, and let $k_{\max}(s)$ denote the largest positive integer $k$ for which $F_{s,k}$ has an integer root. $k_{\max}$-conjecture. For all $s>3$,

$$k_{\max}(s)=2s-1.$$

Equivalently, the polynomial $F_{s,k}$ can have integer roots only when $k\leqslant 2s-1$. This is presented as the most important open question about the roots of Moser polynomials; its resolution would bound the values of $k$ that need to be considered when studying singular pairs.