The rotational case of the $\Lambda\Sigma=\Delta$ conjecture

Let $W$ be the Coxeter group in the paper, with Coxeter number $h$, rank $n$, and identity $1_W$. Let $\mathbf{R}$ generate the rotation subgroup $\operatorname{Rot}_{h+2}$, and let $\Lambda$ be the character on the relevant reduced homology representation. If $\mathbf{F}$ is a power of $\mathbf{R}$ and $\mu$ is a root of unity having the same order as $\mathbf{F}$, then $\ker(w-\mu 1_W)$ is the eigenspace of $w$ with eigenvalue $\mu$.

Rotational $\Lambda\Sigma=\Delta$ conjecture.

$$\Lambda(\mathbf{F},w)=(-1)^n\cdot(-h-1)^{\dim\ker(w-\mu 1_W)}.$$

This is the rotation-specialized reformulation of the main character conjecture. The supplied text does not state whether this specialization has been resolved independently.