The topological and smooth Hausmann-Weinberger invariants disagree

For $\pi$ a finitely presented group, Hausmann and Weinberger defined $q(\pi )\in \mathbb{Z}$ to be the minimum Euler characteristic over all closed, oriented $4$-manifolds with fundamental group $\pi$. This short note establishes that this minimum value in general differs depending on whether one minimizes over topological manifolds or only those admitting a smooth structure.