A q-Narayana positivity conjecture

Let $n$ and $r$ be positive integers, and let $a$ be a non-negative integer. Define the $q$-Narayana numbers by

$$N_q(n,k)=\frac{1}{[n]}{n\brack k}{n\brack k-1}.$$

Let $E_{n,r,a,j}(q)$ denote the Laurent polynomial expression in the preceding corollary. The q-Narayana conjecture. For every integer $j$, the preceding corollary remains valid, and $E_{n,r,a,j}(q)$ has non-negative integer coefficients when $0\leqslant j\leqslant 2r$. This conjecture extends the proved range for the corresponding q-Narayana divisibility statement and is proposed as a further generalization.