Smith–Thom inequality for topological groupoids with involution

Let ${\mathscr{X}}=[R\rightrightarrows U]$ be a topological groupoid with involution $\sigma\colon {\mathscr{X}}\rightarrow {\mathscr{X}}$. Assume that $R$ and $U$ are locally compact and Hausdorff, and that the spaces $\left|{\mathscr{X}}^{G}\right|$ and $\left|{\mathcal{I}}_{{\mathscr{X}}}\right|$ have finite-dimensional $\mathbb{Z}/2$-cohomology.

Smith–Thom conjecture. The inequality

\textnormal{\dim} {\mathrm{H}}^\ast\left(\left|{\mathscr{X}}^{G}\right|,\mathbb{Z}/2\right)\leq \textnormal{\dim} {\mathrm{H}}^\ast\left(\left|{\mathcal{I}}_{{\mathscr{X}}}\right|,\mathbb{Z}/2\right)

should hold. This generalizes the usual Smith–Thom inequality from topological spaces to topological groupoids with involution. When the groupoid comes from a topological space, the inertia realization is homeomorphic to the space itself, and the conjectural inequality reduces to the classical one; the proposed bound is also sharp in examples of groupoids with nontrivial automorphisms.