This is a follow-up to my previous question, but asks a much simpler one:
Let $cd$ denote the cohomological dimension of a group, i.e. the minimal length of a projective resolution of $\mathbb{Z}$ over the group ring.
Is is true for every group $G$ of finite cohomological dimension that $$cd(G \times G) \stackrel{?}{>} cd(G )$$