The non-negative coefficient conjecture for cylindric partition generating functions

Let $c=(c_1,c_2,c_3)$ be a profile, let $P_{n,c}(q)$ be the polynomial defined by

$$f_{n,c}(q)=\frac{P_{n,c}(q)}{(q^3;q^3)_n},$$

and let $g_{n,c}=[z^n]G_c(z,q)$. Write $\ell=c_1+c_2+c_3$.

Non-negative coefficient conjecture. For any profile $c=(c_1,c_2,c_3)$, the polynomial $P_{n,c}(q)$ has non-negative coefficients. Moreover, if $\ell\not\equiv 0\pmod{3}$, then

$$g_{n,c}=\frac{Q_{n,c}(q)}{(q;q)_n},$$

where $Q_{n,c}(q)$ is a polynomial in $q$ with non-negative coefficients and

$$Q_{n,c}(1)=\left(\frac{(\ell+2)(\ell+1)}{6}-1\right)^n.$$

This conjecture seeks a positive-coefficient description of generating functions for cylindric partitions beyond the cases established in the paper; the general sum-side structure remains unknown, and the authors state that they do not yet have tools to determine it for arbitrary profiles.