Weak K-moduli folklore conjecture

Let $\mathcal{M}$ be a moduli algebraic stack of polarized log-terminal Calabi–Yau varieties with quasi-projective coarse moduli variety $M$. A weak K-moduli stack is a proper Deligne–Mumford compactification equipped with a suitable $\mathbb{Q}$-Gorenstein family of polarized semi-log-canonical Calabi–Yau varieties extending the universal family, with effectiveness of the underlying family in the weak, but not very weak, case. Weak K-moduli folklore conjecture. Every such $\mathcal{M}$ admits at least one weak K-moduli proper stack. If $\mathcal{M}$ is uniformized by a Hermitian symmetric domain, then the normalization of at least one such compactification is dominated by a toroidal or semi-toric compactification. This expectation is motivated by the known toroidal and semi-toric compactifications for principally polarized abelian varieties and polarized K3 surfaces; its general validity remains open.