The extended Stanley inequality for minimum positions of subsets

Let $P=(X,\prec)$ be a poset with $|X|=n$ elements. Fix a nonempty subset $A\subseteq X$, and let $f$ be chosen uniformly from the set of linear extensions of $P$. Write $f_{\min}(A)=\min\{f(x):x\in A\}$. Extended Stanley inequality. For $2\leq k\leq n-1$,

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

This extends Stanley's log-concavity inequality from the position of a single element to the minimum position attained on an arbitrary nonempty subset. The source presents it as a proposed extension; its resolution is not specified here.