Non-negative reduced series conjecture for coloured plane partitions

Let $G=G(a,b,c)$ be a finite abelian subgroup of ${\rm SL}_3(\mathbb C)$ of order $r$, diagonally embedded, with $\mathbb C^3\cong\rho_a\oplus\rho_b\oplus\rho_c$. Let $Z^{G(a,b,c)}(\mathbf{t})$ be the generating function of $G(a,b,c)$-coloured plane partitions, and let $M(t)$ be the generating function of all plane partitions. Set

$$t=\prod_{j=0}^{r-1}t_j.$$

Reduced-series conjecture. There exists a factorisation

$$Z^{G(a,b,c)}(\mathbf{t})=M(t)^r\cdot Z^{G(a,b,c)}_{\rm red}(\mathbf{t}),$$

where the reduced series $Z^{G(a,b,c)}_{\rm red}(\mathbf{t})$ has only non-negative coefficients.

This conjecture predicts positivity after removing the universal plane-partition contribution $M(t)^r$. It is motivated by the analogy with coloured partitions and by geometric ideas developed later in the paper; no resolution is supplied here.