Fix a finitely generated Coxeter system $(W, S)$, and let $W_J$ denote the standard parabolic proper subgroup generated by a subset $J \subset S$. It is well known that the poset of cosets $\{xW_J\}$ ordered by reverse inclusion is face poset of Coxeter complex.

Consider the poset of cosets $\{xH\}$, ordered by reverse inclusion, $H$ run over proper reflection subgroup of $W$(subgroup generated by a collection of reflections in $W$). This poset may not be face poset of a complex. Do we know any topological property of this poset, is this poset spherical ?