Strong Sperner conjecture for weak orders on finite Coxeter groups

Let $C$ be a finite Coxeter group, equipped with its weak order. A ranked poset is strongly Sperner if, for every positive integer $r$, the maximum size of a union of $r$ antichains equals the sum of the $r$ largest rank sizes. Strong Sperner conjecture. The weak order on any finite Coxeter group is strongly Sperner. This would extend the known strong Sperner property of weak order for the symmetric group to all finite Coxeter groups; the claim remains open in the source.