Boundedness conjecture for complements

Let ${\EuScript{N}}_d({\Phi})$ denote the set of positive integers occurring as complement indices in the setting of the assumptions referred to in the source. Boundedness conjecture for complements. The set ${\EuScript{N}}_d({\Phi})$ is finite. This conjecture is known in dimension two, where its proof relies on boundedness results for log del Pezzo surfaces; in arbitrary dimension it remains open.