Bredon cohomological dimension for virtually abelian stabilisers for CAT(0) groups

Given a discrete group $G$, for any integer $r⩾0$ we consider the family of all virtually abelian subgroups of $G$ of rank at most $r$. We give an upper bound for the Bredon cohomological dimension of $G$ for this family for a certain class of groups acting on $\mathrm{C}\mathrm{A}\mathrm{T}(0)$ spaces. This covers the case of Coxeter groups, Right-angled Artin groups, fundamental groups of special cube complexes and graph products of finite groups. Our construction partially answers a question of J.-F. Lafont.