Kawamata–Matsuki finiteness conjecture for wlc klt models

Let $(X,B)$ be a projective wlc model, and let a $0$-class denote the equivalence class relevant to the fixed wlc pair in the source. A projective wlc klt model is a projective model with the stated weak log canonical and klt structure, and two such models are log isomorphic when they are isomorphic as log pairs.

Finiteness of minimal models. The number of (log) isomorphism classes of projective wlc klt models in a fixed $0$-class is finite.

This is the log-pair generalization of the Kawamata–Matsuki finiteness conjecture. The source notes that it is known for wlc models of general type and for log surfaces with non-zero boundary, while the original finiteness problem remains open in dimension at least $3$.