Conjecture on possible rational roots of alternating sign triangle weight functions

Let $M$ be an irreducible Motzkin path of length $n\geq 8$, and let $S$ be an irreducible centred Catalan set of size $n\geq 11$. The rational roots of the associated weight functions are considered in the following two cases.

Possible-roots conjecture. For $M$, the rational roots of $w_l(M)$ lie in

$$\{-1,-2,\ldots,-2n+2\},$$

and every integer in this set is a root of $w_l(M)$ for some Motzkin path $M$. For $S$, the rational roots of $w_l(S)$ lie in

$$\{-1,\ldots,-2n+4,-\frac{n^2-5n+7}{n-3}\},$$

and every integer in this set is a root of $w_l(S)$ for some centred Catalan set $S$. Moreover,

$$-\frac{n^2-5n+7}{n-3}$$

is a root of $w_l(S)$ if and only if $S=\{-n+2,-1,0,1,\ldots,n-3\}$ or $S=\{-n+3,\ldots,-1,0,1,n-2\}$.

The conjecture seeks to determine which rational numbers occur as roots of the weight functions. The source presents it as a data-driven conjecture, with a broader goal of describing all rational roots of $w_l(S)$ and $w_l(M)$; no resolution is supplied here.