The ideal average-height conjecture for posets

Let $P$ be a finite poset, let $A$ be an ideal of $P$, and let $h(x)$ denote the expected position of $x$ in a uniformly random linear extension. Write $\max(A)$ for the maximal elements of $A$. The ideal average-height conjecture.

$$\max_{x\in A}h(x)\geq |A|-|\max(A)|+1.$$

The conjecture would imply the bound $\operatorname{gap}(P)\leq 2w(P)-1$. It is true and best possible when $A=P$, but the general case remains open, even for width two.