An Eilenberg–Ganea phenomenon for actions with virtually cyclic stabilisers

In dimension 3 and above, Bredon cohomology gives an accurate purely algebraic description of the minimal dimension of the classifying space for actions of a group with stabilisers in any given family of subgroups. For some Coxeter groups and the family of virtually cyclic subgroups we show that the Bredon cohomological dimension is 2 while the Bredon geometric dimension is 3.