history

1. in [2], Kollár proved the conjecture for Q-factorial germs of log canonical pairs; 2. in dimension two, McKernan and Keel proved the conjecture for projective surfaces of Picard rank one, [3]; Shokurov [1] proved the conjecture for arbitrary morphisms of surfaces; 3. in an unpublished note, Chelstov proved the conjecture for Q-factorial projective varieties of Picard number 1; 4. in [4], Prokhorov proved the conjecture for certain projective 3-folds. The method of his proof relies on the minimal model program in dimension three; 5. in [5], Yao gives a proof of the above conjecture for log smooth projective pairs (X, D) with KX +D ∼Q 0. Yao’s proof was inspired by the mirror-symmetry techniques of [6]; 6. finally, in [7], the authors settled the conjecture in the non-relative case, i.e., for a log canonical pair (X, B), where X is a proper variety - this setup corresponds to the relative setup where Z is just a point. (page) (Moraga and Svaldi, 2025, p. 3)