Felsner–Krawczyk–Micek conjecture for poset antichains

For a positive integer $k$, let $\mathbf{k}+\mathbf{k}$ be the poset consisting of two disjoint $k$-element chains with no comparabilities between distinct chains. Let $\mathcal{P}(k)$ be the class of posets containing no subposet isomorphic to $\mathbf{k}+\mathbf{k}$, and let $\mathcal{P}(k,w)$ be the subclass consisting of posets of width at most $w$. For a poset $P$, write $\mathcal{M}(P)$ for its family of maximum antichains. Felsner–Krawczyk–Micek conjecture. For positive integers $k,w$ with $k\ge 2$, the maximum width of $\mathcal{M}(P)$ over posets $P\in\mathcal{P}(k,w)$ is

$$(k-1)^{w-1}.$$

This conjecture is an equivalent poset formulation of the crossing-vector extremal problem, using the bound relating the width of the family of maximum antichains to $f(k-1,w)$. Its general status is not resolved in the supplied text.