The continuous window–variance conjecture for posets

Let $P$ be a finite poset, let $F$ be uniformly distributed in its order polytope, let $H(x)=\mathbb E F(x)$, and let $\operatorname{Win}(x)=\mathbb E[R(x)-Q(x)]$, where $Q(x)=\max_{y<x}F(y)$ and $R(x)=\min_{y>x}F(y)$ with endpoint conventions $Q(x)=0$ and $R(x)=1$. The window–variance conjecture. For every $P$ and $x\in P$,

$$\operatorname{Var}(F(x))\leq \frac{\operatorname{Win}(x)H(x)(1-H(x))}{2+\operatorname{Win}(x)}.$$

The proposed bound is tight in a stated incomparability case and is known for series-parallel posets, but remains open in general.