Infinite q-log-convexity conjecture for longest-increasing-subsequence polynomials

For each $n\geq 1$, let $P_n(x)=\sum_k P_{n,k}x^k$, where $P_{n,k}$ is the number of permutations of $[n]=\{1,2,\ldots,n\}$ whose longest increasing subsequence has length $k$. A polynomial sequence is infinitely q-log-convex when every iterate of the relevant $q$-log-convexity operator is $q$-log-convex. Infinite $q$-log-convexity conjecture. The polynomial sequence $\{P_n(x)\}$ is infinitely $q$-log-convex. This proposes an iterated $q$-analogue of log-convexity for longest-increasing-subsequence polynomials; the supplied text gives no resolution status.