Chen's log-concavity conjecture for longest increasing subsequence counts

Let $\mathcal{S}_n$ be the symmetric group on $[n]=\{1,2,\dots,n\}$. For $\sigma\in\mathcal{S}_n$, let $\ell_n(\sigma)$ be the length of its longest increasing subsequence, and define

$$L_{n,k}=\{\sigma\in\mathcal{S}_n:\ell_n(\sigma)=k\},\qquad \ell_{n,k}=|L_{n,k}|.$$

Chen's conjecture. For any fixed $n$, the sequence $\ell_{n,1},\ell_{n,2},\ldots,\ell_{n,n}$ is log-concave. Equivalently, the distribution of $\ell_n(\sigma)$ for a uniformly chosen random permutation $\sigma$ is log-concave.

This conjecture concerns the shape of the longest-increasing-subsequence distribution in a uniformly random permutation and is related to the extensive asymptotic theory of that statistic. The supplied text attributes the conjecture to Chen and refers to further discussion by Bona, Leader, and Lee, but gives no resolution status.