The presentation conjecture for invariant rank-two parabolic arrangements

Let $W$ be a finite real reflection group, and let $\mathscr{P}$ be a collection of rank $2$ parabolic subgroups of $W$ closed under conjugation. Set

$$\mathscr{W}=\{\operatorname{Fix}(G):G\in\mathscr{P}\}.$$

Define a Coxeter group $W'$ with the same generating set $S$ as $W$, with relations determined by

$$m'(s,t)=\begin{cases}\infty&\text{if }\langle s,t\rangle\in\mathscr{P},\m(s,t)&\text{otherwise},\end{cases}$$

and let $\varphi':W'\to W$ send $s$ to $s$ for every $s\in S$. The presentation conjecture. The fundamental group of the complement satisfies

$$\pi_1(\mathcal{M}(\mathscr{W}))\cong\ker\varphi',$$

and $\mathscr{W}$ is a $K(\pi,1)$-arrangement, where $\pi=\ker\varphi'$. This would give a presentation of the fundamental group for complements of invariant codimension-two subspace arrangements; the conjecture is presented as an open question following the known $K(\pi,1)$ results for the relevant arrangements.