q-Delannoy positivity conjecture

For positive integers $m,n,r$, let $D_q(m,k)$ and $D_{q^{-1}}(m,k)$ be the natural $q$-Delannoy numbers defined by

$$D_q(m,n)=\sum_{j=0}^{n}q^{\binom{j}{2}}{n\brack j}{n+m-j\brack n},$$

with $D_{q^{-1}}(m,n)=q^{-mn}\sum_{j=0}^{n}q^{\binom{j+1}{2}}{n\brack j}{n+m-j\brack n}$. q-Delannoy positivity conjecture. Each of the following is a Laurent polynomial in $q$ with non-negative integer coefficients:

$$\sum_{k=0}^{n-1}\frac{(1-q^m)(1-q^{m+1})(1-q^{2k+1})}{(1-q^2)(1-q^n)^2}D_q(m,k)D_{q^{-1}}(m,k)q^{-k},$$ $$\sum_{k=0}^{n-1}\frac{1-q^{2k+1}}{1-q^n}D_q(m,k)^rD_{q^{-1}}(m,k)^rq^{-k},$$

and

$$\sum_{k=0}^{n-1}(-1)^{n-k-1}\frac{1-q^{2k+1}}{1-q^n}D_q(m,k)^rD_{q^{-1}}(m,k)^rq^{\binom{k}{2}}.$$

The claim extends the paper's integer-valued polynomial phenomena to a proposed $q$-analogue. It is left open because the authors lack a sufficiently useful single-sum expression for the product of the two $q$-Delannoy numbers.