Saks's extremal gap conjecture for heights of incomparable elements

For a poset $P$, let $h(P,x)$ denote the average height of $x$, and let $\eta(P)$ be the smallest absolute difference $|h(P,x)-h(P,y)|$ over incomparable elements $x,y$. Set

$$\vartheta=\frac14\prod_{k=1}^{\infty}\left(1-\frac1{2^k}\right)^{-1}.$$

Saks’s gap conjecture. Every poset $P$ that is not a chain satisfies

$$\eta(P)\leq\vartheta.$$

The source gives constructions approaching $\vartheta$ and reports the conjecture as open.