Let $G$ be a virtually compact special group, in the terminology of Haglund and Wise (i.e., $G$ has a finite index subgroup $H$ which is isomorphic to the fundamental group of some compact special non-positively curved cube complex $X$).
Question: for any element $g \in G$, is the centralizer $C_G(g)$ also a virtually compact special group?
From the work of Haglund and Wise we know that the answer is positive if $G$ is hyperbolic, so I am actually asking about the non-hyperbolic case.
In general, some properties of $C_G(g)$ can be deduced from the fact that $G$ is semihyperbolic (in particular, the CAT$(0)$-quasiconvexity), but I am not sure whether these are helpful for the question.