Unimodality and log-concavity of involution length profiles in classical Coxeter groups

Let $W(A_n)$, $W(B_n)$, and $W(D_n)$ denote the finite Coxeter groups of the indicated types. For each group, consider the even involution length profile and the odd involution length profile, namely the sequences counting involutions according to their even or odd length. A sequence $(x_i)_{i=1}^N$ is log-concave if $x_i^2\geq x_{i-1}x_{i+1}$ for $2\leq i\leq N-1$, and unimodal if its terms first weakly increase and then weakly decrease.

Unimodality and log-concavity conjecture. The even and odd involution length profiles of $W(A_n)$ are log-concave, while the even and odd involution length profiles of $W(B_n)$ and $W(D_n)$ are unimodal.

The conjecture concerns coefficient profiles of involution length polynomials in the classical finite Coxeter groups. The paper reports verification for the classical groups through rank $10$ and for all exceptional groups, but also explains that the asserted unimodality for $W(B_n)$ is false in general; consequently, the combined conjecture as stated is refuted.