Characteristic-independence conjecture for formal characters of Khovanov–Lauda–Rouquier modules

Let $(I,\cdot)$ be a Cartan datum of finite type, let $Q_+$ be its positive root lattice, let $\alpha\in Q_+$, and let $R_\alpha$ be the corresponding Khovanov–Lauda–Rouquier algebra over a ground field $\mathbb{F}$. The formal character of a graded $R_\alpha$-module records its graded weight multiplicities. Characteristic-independence conjecture. The formal characters of the irreducible graded $R_\alpha$-modules are independent of the characteristic of the ground field $\mathbb{F}$. This is asserted only for Cartan data of finite type; the analogous statement is known to fail in affine type $A$, where the James conjecture gives a more subtle characteristic-dependent picture.