The conjecture that large incomparability degree forces asymptotic balance

Let $P$ be a finite poset. For $x\in P$, let $\Pi(x)=\\{y\in P\setminus\\{x\\}:y\not\sim x\\}$, let $\pi(x)=|\Pi(x)|$, and let $\pi(P)=\max_{x\in P}\pi(x)$. Let $\delta(P)$ denote the maximum balance parameter over distinct pairs in $P$. The incomparability-degree conjecture. If

$$\pi(P)\longrightarrow\infty,$$

then $\delta(P)\to 1/2$. This conjecture is presented as an old conjecture of the second author and would imply the Kahn–Saks conjecture. It is known for every fixed width, but remains open without a fixed-width restriction.