Unimodular Frostman measure equality conjecture

Let $\boldsymbol{D}$ be the random metric object in the unimodular Frostman setting, let $\mathcal H^{\alpha}_{M}(\boldsymbol{D})$ denote its unimodular Hausdorff measure, and let $\xi^{\alpha}_M(1)$ be the corresponding Frostman quantity evaluated at the constant function $1$. Unimodular Frostman equality conjecture. One has

$$\mathcal H^{\alpha}_{M}(\boldsymbol{D})=\xi^{\alpha}_M(1).$$

The text notes that this equality holds for compact continuum spaces, while for point-stationary point processes in $\mathbb R^k$ only an upper bound with a dimension-dependent constant has been shown; thus the general assertion remains open.