Narayana polynomials form a q-log-convex sequence

The Narayana number $N(n,k)$ counts Dyck paths of length $2n$ with exactly $k$ peaks, and the Narayana polynomials are $N_n(q)=\sum_{k=0}^{n}N(n,k)q^k$. A sequence of polynomials is q-log-convex when each difference $N_{n+1}(q)N_{n-1}(q)-N_n(q)^2$ has nonnegative coefficients. Narayana polynomials conjecture. The Narayana polynomials $N_n(q)$ form a $q$-log-convex sequence. This was verified for $n\leq 100$ using Maple; pointwise log-convexity for each fixed nonnegative $q$ is known, but coefficientwise $q$-log-convexity remains the conjectural strengthening.