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

For integers $d,r\geqslant1$, let $c=(c_0,\dots,c_{r-1})$ be a profile of level $d$, meaning

$$c_0+\cdots+c_{r-1}=d.$$

Let $Q_{n,c}(q)$ be the formal power series defined from the cylindric-partition generating function by the source's formula. Arbitrary-rank cylindric-partition positivity conjecture. The series $Q_{n,c}(q)$ is a polynomial in $q$ with nonnegative coefficients and

$$Q_{n,c}(1)=\left(\frac{\gcd(d,r)}{d+r}\binom{d+r}{r}-\gcd(d,r)\right)^n.$$

This generalises the rank-$3$ conjecture of Corteel, Dousse and Uncu and is motivated by the expected interpolation between finite-rank Andrews--Gordon identities; the supplied context does not state a proof.