The local deletion sensitivity conjecture for posets

Let $P$ be a finite poset with $n$ elements, let $x,y\in P$ with $y\ne x$, and let $h_P(y)$ denote the expected position of $y$ in a uniformly random linear extension of $P$. The local deletion sensitivity conjecture. The bound

$$|h_P(y)-h_{P-x}(y)|=O(1)$$

holds whenever any one of the following conditions holds: (a) there is a chain $x=x_0<x_1<\cdots<x_k=y$ with $k=\Omega(n)$; (b) the distance between $x$ and $y$ in the cover graph of $P$ is $\Omega(n)$; or (c) $\pi(x)=O(1)$. The paper notes that the weaker condition $\operatorname{win}(x)=O(1)$ might suffice in (c); the conjecture itself is open.