Brenti–Carnevale–Tenner's rank-symmetry conjecture for odd diagram classes

Let $S_n$ be the symmetric group on $[n]$. For $w\in S_n$, define its odd diagram by

$$D_o(w)=\{(i,j)\colon w(i)>j,\ i<w^{-1}(j),\ i\not\equiv w^{-1}(j) \pmod 2\}.$$

An odd diagram is a set $D=D_o(w)$, and its odd diagram class is

$$\mathrm{Perm}(D)=\{w\in S_n\colon D_o(w)=D\}.$$

Brenti–Carnevale–Tenner's conjecture. Each odd diagram class is rank-symmetric in the Bruhat order.

Odd diagram classes are known to be intervals in the Bruhat order. The conjecture asserts the stronger symmetry property that the ranks in each such interval are symmetric; the paper's abstract states that its results resolve this conjecture by factoring the associated Poincaré polynomial into factors of the form $1+t+\cdots+t^m$.