Finiteness conjecture for projective weakly log canonical klt models

Let a 0-class be a class of crepant birationally equivalent log pairs, and let a projective wlc klt model be a projective weakly log canonical model with klt singularities in that class.

Finiteness conjecture for wlc klt models. The number of projective wlc klt models in a fixed 0-class is finite up to log isomorphism.

This is a log-pair extension of the finiteness problem for minimal models. The paper’s abstract states that it proves finiteness for log surfaces by reducing the problem to boundedness of the polarization, while the supplied text does not specify the precise general status of this conjecture.