Infinite log-concavity 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 is infinitely log-concave when every polynomial obtained by iterating the log-concavity operator remains log-concave. Infinite log-concavity conjecture. $P_n(x)$ is infinitely log-concave for $n\geq 1$. This strengthens the preceding log-concavity proposal for the longest-increasing-subsequence distribution; the supplied text gives no resolution status.