James conjecture for affine Khovanov–Lauda–Rouquier algebras

Let $Q_+$ be the positive root lattice, let $\boldsymbol{\Lambda}$ be a dominant weight, and let $R_\alpha^{\boldsymbol{\Lambda}}(K)$ be the corresponding cyclotomic quotient of the affine Khovanov–Lauda–Rouquier algebra $R_\alpha(K)$. For an irreducible $R_\alpha(K)$-module $L_K$ factoring through this quotient, reduction modulo a prime $p$ is formed from an integral invariant lattice.

James conjecture. Let $\alpha\in Q_+$, and let $L_K$ be an irreducible $R_\alpha(K)$-module which factors through $R_\alpha^{\boldsymbol{\Lambda}}(K)$. Then reduction modulo $p$ of $L_K$ is irreducible provided

$$p>(\boldsymbol{\Lambda},\alpha)-(\alpha,\alpha)/2.$$

This extends the block version of the classical James conjecture to affine type; for affine type ${\tt A}_l^{(1)}$ with $\boldsymbol{\Lambda}=\boldsymbol{\Lambda}_0$, it is equivalent to that block conjecture. The stated bound is motivated by earlier results, but no resolution is supplied here.