Kleshchev–Ram conjecture on decomposition numbers for Dynkin KLR algebras

Let Khovanov–Lauda–Rouquier algebras be associated to Dynkin quivers, and consider their decomposition numbers, which record the graded composition multiplicities of their modules.

Kleshchev–Ram conjecture. The decomposition numbers for KLR algebras associated to Dynkin quivers are trivial.

This was proposed as a finite analogue of the James conjecture, motivated by the hope that the modular representation theory of KLR algebras for Dynkin quivers would be simpler than the cyclic-quiver case. The source does not state whether the conjecture has been resolved; it is therefore recorded as open.