  We give a characterization of toroidal (resp., semi-toric) compactifications
due to Ash-Mumford-Rapoport-Tai (resp., Looijenga) as log minimal models and
apply it to study weak K-moduli compactifications, giving a different proof to
a theorem of Alexeev-Engel.
  We also discuss towards further generalization, in particular revisit
Shah-Sterk compactification of moduli of polarized Enriques surfaces to show
compatibility with log K-stability.
