Bóna–Lackner–Sagan's log-concavity conjecture for involutions

Let $n$ be a positive integer. For $1\leq k\leq n$, let $I_{n,k}$ be the set of involutions in the symmetric group $\mathfrak{S}_n$ whose longest increasing subsequence has length $k$, and let $i_{n,k}=|I_{n,k}|$. Bóna–Lackner–Sagan's conjecture. For any fixed $n$, the sequence $\{i_{n,k}\}_{k=1}^n$ is log-concave. This is the involution analogue of the permutation conjecture; the paper studies a stronger Schur-positivity formulation, but does not prove the stated conjecture in general.