Minimal skeleton conjecture for stacky dlt compactifications

Fix a (Kawamata-)log terminal Deligne–Mumford stack $\mathcal{U}$, and let $(\mathcal{X},\sum_{1\le i\le s}\mathcal{D}_{i})$ be a stacky dlt model such that

$$\mathcal{X}\setminus\sum_{i}\mathcal{D}_{i}=\mathcal{U}.$$

Minimal skeleton conjecture. The homeomorphism type of the dual complex of $(\mathcal{X},\sum_{1\le i\le s}\mathcal{D}_{i})$ does not depend on the choice of such compactification.

This asserts that the dual complex associated with a minimal stacky dlt compactification is an invariant of the fixed log terminal Deligne–Mumford stack. Versions of the corresponding statement are known in the schematic case, while the stacky formulation remains conjectural here.