  We give a counterexample to the most optimistic analogue (due to Kleshchev
and Ram) of the James conjecture for Khovanov-Lauda-Rouquier algebras
associated to simply-laced Dynkin diagrams. The first counterexample occurs in
type A_5 for p = 2 and involves the same singularity used by Kashiwara and
Saito to show the reducibility of the characteristic variety of an intersection
cohomology D-module on a quiver variety. Using recent results of Polo one can
give counterexamples in type A in all characteristics.
