Hurwitz stability conjecture for Laguerre-polynomial resultants

Let $P_n(s,z)$ denote the polynomials considered in the paper, and let $\operatorname{res}_z$ denote the resultant with respect to $z$. A polynomial is stable when all its roots have negative real part. For positive integers $n,m$ with $n\leq m$ and a real parameter $a>0$, consider

$$\operatorname{res}_z\bigl(P_n(s,z),P_m(s+a,z)\bigr).$$

Resultant stability conjecture. For any positive integers $n,m$ such that $n\leq m$, and any $a>0$, the resultant $\operatorname{res}_z(P_n(s,z),P_m(s+a,z))$ is stable.

When $n=m$ and $a=1$, this reduces to the conjectured stability of $\theta_n(s)$. The claim is motivated by numerical experiments and by the relation between $\theta_n(s)$ and the resultant; the paper notes that even the case $n=1$ appears difficult, where the assertion becomes stability of $P_m(s+a,-s)$ for every $a>0$.