Positivity conjecture for alternating sums of q-Narayana numbers

Let $n$ and $r$ be positive integers, let $0\leqslant j\leqslant 2r-1$, and let $C_n(q)$ and $N_q(2n+1,n+k+1)$ denote the q-Catalan and q-Narayana quantities used in the paper. Positivity conjecture. The expression

$$\frac{1}{C_n(q)}\sum_{k=-n}^{n}(-1)^k q^{jk^2+{k\choose 2}}N_q(2n+1,n+k+1)^r$$

is a polynomial in $q$ with nonnegative integer coefficients. Numerical calculation implies that this conjecture fails when $j\geqslant 2r$, so the stated range $0\leqslant j\leqslant 2r-1$ is essential.