Kirillov's eventual polynomiality conjecture for stretched Schubert coefficients

Let $u,v,w\in S_n$. For $N\geq 1$, let $f_{u,v,w}(N)$ denote the stretched Schubert structure coefficient defined in the paper.

Kirillov's conjecture. $f_{u,v,w}(N)$ is eventually polynomial in $N$.

Kirillov's conjecture is equivalent to the generating function $P_{u,v,w}(t)$ having no pole other than $t=1$. The paper proves only eventual quasi-polynomiality, equivalently that all poles of $P_{u,v,w}(t)$ are roots of unity, so the stronger polynomiality statement remains open.