Smith–Thom inequality for real Deligne–Mumford stacks

Let $\mathcal{X}$ be a separated Deligne–Mumford stack of finite type over $\mathbb{R}$, and let $I_{\mathcal{X}}$ be the coarse moduli space of its inertia stack.

Real stack Smith–Thom conjecture. The dimension of the mod-$2$ cohomology of the real locus should satisfy

\textnormal{\dim} {\mathrm{H}}^\ast\left(\left|\mathcal{X}(\mathbb{R})\right|,\mathbb{Z}/2\right)\leq \textnormal{\dim} {\mathrm{H}}^\ast\left(I_{\mathcal{X}}(\mathbb{C}),\mathbb{Z}/2\right).

This is presented as the algebraic version obtained by restricting the topological groupoid conjecture to groupoids arising from real Deligne–Mumford stacks. Its status is not resolved in the supplied text.