On boundaries of parabolic subgroups of Coxeter groups

In this paper, we investigate boundaries of parabolic subgroups of Coxeter groups. Let $(W,S)$ be a Coxeter system and let $T$ be a subset of $S$ such that the parabolic subgroup $W_T$ is infinite. Then we show that if a certain set is quasi-dense in $W$, then $W \partial\Sigma(W_T,T)$ is dense in the boundary $\partial\Sigma(W,S)$ of the Coxeter system $(W,S)$, where $\partial\Sigma(W_T,T)$ is the boundary of $(W_T,T)$.