The rank log-concavity conjecture for Weyl-group Bruhat intervals

Let $W$ be a Weyl group and let $u,v\in W$. The Bruhat interval $[u,v]$ is the ranked poset of elements between $u$ and $v$ in Bruhat order.

Rank log-concavity conjecture. The interval $[u,v]$ is rank log-concave.

The conjecture concerns the rank generating functions of Bruhat intervals. It has been verified in several Weyl-group types and for dihedral groups, while the analogous assertion fails for some finite Coxeter systems, such as type $H_3$; the Weyl-group claim remains open in general.