Nonnegative coefficient conjecture for three-residue partition products

Let $L>0$, $z>1$, $y>z$, $y<x\leq y+z$, and $m\geq yz+2$. For the finite $q$-Pochhammer symbol $(a;q^m)_L=\prod_{j=0}^{L-1}(1-aq^{mj})$, define

$$\frac{1}{(q,q^{x},q^{yz};q^{m})_L}-\frac{1}{(q^z,q^{y},q^{yz+1};q^{m})_L}=\sum_{n=1}^{\infty}a(L,y,n,x,z,m)q^n.$$

Three-residue coefficient conjecture. If $z$ does not divide $y$, then all coefficients $a(L,y,n,x,z,m)$ are non-negative; if $z$ divides $y$, then only finitely many of these coefficients are negative.

This conjecture extends the paper's partition inequalities to three residues and distinguishes the cases according to whether $z$ divides $y$. The authors state that it is motivated by experimental evidence and is being actively pursued; its resolution is not supplied here.