Homological dimension conjecture for virtually cyclic classifying spaces

Let $\Gamma$ be a discrete subgroup of a Lie group that is not virtually cyclic. Write $vcd(\Gamma)$ for its virtual cohomological dimension, and let $B_{\mathcal{VC}}\Gamma$ denote the classifying space for the family of virtually cyclic subgroups, with $hdim$ denoting homological dimension. Homological dimension conjecture. One should have

$$hdim(B_{\mathcal{VC}}\Gamma)=vcd(\Gamma)+1.$$

This question is stated as unsolved in the paper and extends the dimension computation proved there for crystallographic groups.