The locally conical reflection arrangement conjecture

Let $W$ be a well-generated reflection group acting on a complex vector space $V$. A generating set $R$ is well-generating when it generates $W$ and has cardinality $\dim V$. The system $(W,R)$ is called abstractly locally conical if, for every $U,T\subseteq R$ with $U$ nonempty, every Quillen fiber of

$$\mathrm{Supp}:\mathrm{Face}(\Delta^U_T)\to \Pi^U_T\setminus\{\hat{1}\}$$

has a cone point. The locally conical reflection arrangement conjecture. For each well-generated reflection group $W$, there exists a well-generating $R$ for which $(W,R)$ is abstractly locally conical. This property would provide the local conical structure needed to compare the relevant simplicial complexes with the intersection lattice of the reflection arrangement; whether every well-generated reflection group admits such a generating set is left open.