Brown's conjecture on the geometric and virtual cohomological dimensions of groups

Let $G$ be a virtually torsion-free group with $\operatorname{vcd}G<\infty$. Brown's conjecture has the following weak and strong forms:

Brown's conjecture.

1. In the weak form, there is a contractible proper $G$-CW-complex of dimension $\operatorname{vcd}G$.
2. In the strong form, $\underline{\operatorname{gd}}G=\operatorname{vcd}G$.

The weak form asserts the existence of a proper contractible model of the expected dimension, while the strong form identifies the proper geometric dimension with virtual cohomological dimension. The source presents these as Brown's conjecture and discusses new counterexamples to its strong form; therefore the strong form is refuted, while the displayed weak form is not resolved by the supplied context.