The gap-to-entropy implication for posets

Let $P$ be a finite poset, let $h(x)=\mathbb E f(x)$ for a uniformly random linear extension $f$ of $P$, and let $\operatorname{gap}(P)$ be the maximum consecutive gap among the ordered values $h(x)$. Write $\gamma\leadsto\vartheta$ when $\vartheta$ tends to infinity whenever $\gamma$ does. The gap-to-entropy conjecture.

$$\operatorname{gap}\leadsto\tau,$$

where $\tau(P)=\max_{X\subseteq P}|X|^{-1}H(\varsigma|_X)$ and $H$ is binary entropy. This is the only unresolved implication in the parameter hierarchy displayed in the paper; its status is open.