Corteel--Dousse--Uncu positivity conjecture for cylindric partitions

For a nonnegative integer $n$ and a profile $c=(c_0,c_1,c_2)$, define $d:=c_0+c_1+c_2$, $\ell:=\gcd(d,3)$, and

$$Q_{n,c}(q):=(q^{\ell};q^{\ell})_n[z^n]\bigl((zq)_\infty\operatorname{GK}_c(z,q)\bigr)\in\mathbb Z[[q]].$$

Assume $d\not\equiv0\pmod 3$. Corteel--Dousse--Uncu's conjecture. The series $Q_{n,(c_0,c_1,c_2)}(q)$ is a polynomial in $q$ with nonnegative coefficients and

$$Q_{n,(c_0,c_1,c_2)}(1)=\left(\frac{(d+1)(d+2)}{6}-1\right)^n.$$

Polynomiality and the evaluation at $1$ are known, but positivity remains open.