Dlt log Calabi--Yau compactification conjecture for the Betti moduli space

Let $M_B$ be the Betti moduli space, and let $\mathcal{D}(\partial M_B)$ denote the dual complex associated with the boundary of a dlt compactification. Dlt log Calabi--Yau compactification conjecture. The Betti moduli space $M_B$ admits a dlt log Calabi--Yau compactification $(\overline{M}_B,\partial M_B)$, and the associated dual complex $\mathcal{D}(\partial M_B)$ is homeomorphic to a sphere. Such compactifications refine the expected topological description of the boundary and make the dual complex well-defined even when $M_B$ is singular.