Polynomiality conjecture for multispecies zero-range stationary states

Let $|\bar{P}\rangle$ be the stationary state in a sector $k$, with configurations $(\sigma_1,\ldots,\sigma_L)$ belonging to $\mathcal{B}(k)$, and let $\mathbb{P}(\sigma_1,\ldots,\sigma_L)$ denote its configuration weight. Polynomiality conjecture. For every sector $k$, there is a normalization such that

$$\mathbb{P}(\sigma_1,\ldots,\sigma_L)\in\mathbb{Z}_{\geq 0}[q,-\mu_1,\ldots,-\mu_L]$$

for all $(\sigma_1,\ldots,\sigma_L)\in\mathcal{B}(k)$. This is motivated by the explicit stationary-state formulas and computer experiments; the supplied text gives no resolution, so the conjecture remains open.