Bóna–Lackner–Sagan's log-concavity conjecture for even-column tableaux

Fix a positive integer $n$, and let $\Theta$ be the set of partitions of $n$ all of whose column lengths are even. Let $\ell_{n,k}^{\Theta}$ denote the corresponding quantity associated in the paper with standard Young tableaux and largest part $k$. Bóna–Lackner–Sagan's conjecture. For any fixed $n$, the sequence $\{{\ell}_{n,k}^{\Theta}\}_{k=1}^{n}$ is log-concave. This is presented as a companion to Chen's perfect-matching conjecture; the supplied text gives no resolution, so the conjecture remains open.