Chan–Pak conjecture on the log-concavity of minimum positions in linear extensions

Let $P$ be a finite poset, let $A\subseteq P$, and let $\ell$ be a uniformly random linear extension of $P$. Define

$$\ell_{\min}(A)=\min\{\ell(a):a\in A\}.$$

Chan–Pak conjecture. For every $A\subseteq P$, the sequence of probabilities of the minimum position is log-concave:

$$\P[\ell_{\min}(A)=k]^2 \geq \P[\ell_{\min}(A)=k-1]\cdot\P[\ell_{\min}(A)=k+1].$$

This conjecture generalizes Stanley's log-concavity theorem, recovered when $|A|=1$. The source presents it as a conjecture, and no resolution is supplied here.