Unimodality of length profiles for conjugacy classes of involutions

Let $W(A_n)$ and $W(B_n)$ be finite Coxeter groups, and let $X$ be a conjugacy class of involutions in one of these groups. The even/odd length profile of $X$ is the sequence obtained by counting the elements of $X$ according to their even or odd length. More generally, let $X$ be the set of involutions of odd length in a finite Coxeter group, and consider its odd length profile.

Unimodality conjecture. (i) If $X$ is a conjugacy class of involutions in $W(A_n)$ or $W(B_n)$, then the even/odd length profile of $X$ is unimodal. (ii) If $X$ is the set of involutions of odd length in a finite Coxeter group, then the odd length profile of $X$ is unimodal.

These conjectures are proposed after the preceding global conjecture is shown to fail, while the paper reports supporting computations for classical groups through rank $10$ and all exceptional groups. The source gives no resolution of these revised conjectures.