Positivity rigidity conjecture for a binomial finite-difference sum

Fix $s\in\mathbb{N}^+$, $q\in\mathbb{R}\setminus\{0\}$, $\widetilde{q}\in\mathbb{R}$ with $\widetilde{q}>0$, and real numbers $p_m$ for $m=0,1,\ldots,s+1$. Suppose that, for every $k=0,1,\ldots,s+1$,

$$\sum_{m=0}^{\min(k,s)}\binom{s-1+k-m}{s-1}p_mq^m(-\widetilde{q})^{k-m}\geq 0.$$

Positivity rigidity conjecture. Then $p_0\geq 0$, and for $m=1,2,\ldots,s+1$,

$$p_m=p_0\binom{s}{m}\left(\frac{\widetilde{q}}{q}\right)^m.$$

Moreover, the sum equals $p_0$ for $k=0$ and $0$ for $k=1,2,\ldots,s+1$. The conjecture was verified by Mathematica for $1\leq s\leq 10$ and proved manually for $s=2$, but its general status is unresolved.