Write the basic construction as
$$
U(W,K)=(W\times K)/\sim,
$$
where $(w,x)\sim(w',x)$ exactly when $w^{-1}w'$ belongs to the subgroup generated by the mirrors containing $x$. Let $v_T$ be the vertex of $K$ corresponding to a <spherical subset of a Coxeter system> $T$. Its stabilizer is $W_T$. Therefore
$$
\operatorname{Stab}_W([w,v_T])
=wW_Tw^{-1}.
$$
Every conjugate of every spherical standard parabolic subgroup consequently occurs as a vertex stabilizer.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 111 / 4 / b / Solution

A subset $T\subseteq S$ is spherical when its <standard parabolic subgroup> $W_T$ is finite.

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 Incoming links: Spherical subset of a Coxeter system