Divisibility conjecture for quotient generating series of bow varieties

Let $n,m$ be positive integers and let $e\in\mathbb{Z}^n$ and $f\in\mathbb{Z}^m$ be arbitrary. Write $Z_{\mathrm{quotient}}^c(q,t)$ for the quotient generating series, and interpret divisibility in the sense used in the paper. Divisibility conjecture. The series $Z_{\mathrm{quotient}}^c(q,t)$ is divisible by

$$\prod_{l\geq 1}\frac{1}{(1-t^{2(nl-n+1)}q^l)\cdots(1-t^{2(nl-1)}q^l)(1-t^{2nl}q^l)^{m-1}}.$$

This conjecture predicts a universal product factor in the quotient contribution for arbitrary $n,m,e,$ and $f$; the supplied text gives no proof or resolution.