Panyushev's order conjecture for the positive Panyushev map

Let $W$ be a finite, crystallographic Coxeter group with Coxeter number $h$, let $m$ be the parameter from the preceding theorem, and let ${\sf P}_+$ be the positive Panyushev map on $N\kern-1ptN_+(W)$. Write $\mathcal{R}$ for the set of reflections, $\mathcal{S}$ for the set of simple reflections, $\psi$ for the relevant involution, and $\varepsilon$ for the identity map. Panyushev's conjecture. The order of ${\sf P}_+$ is given by

$${\operatorname{ord}}({\sf P}_+) = \begin{cases} (m+1)h - 2, & \text{if } \psi|_{\mathcal{R} \setminus \mathcal{S}} \text{${}\not\equiv{}$} \varepsilon, \\ (m+1)h/2 - 1, & \text{if } \psi|_{\mathcal{R} \setminus \mathcal{S}} \equiv \varepsilon. \end{cases}$$

This is the positive analogue of Panyushev's conjecture on the order of the Panyushev map, whose non-positive version was proved by Armstrong, Stump, and Thomas. The supplied text does not establish whether this positive version has been resolved.