Branching rule for degenerate cyclotomic Hecke algebras

Grojnowski and Vazirani showed that there is a crystal isomorphism between the crystal defined by modular branching of simple modules of Ariki-Koike algebras and that of an integral highest weight module of the quantum enveloping algebra of $$\widehat{\mathfrak{sl}_e}$$, where $$e$$ is the multiplicative order of the parameter $$q$$. Ariki then showed that this isomorphism is actually the identity, if $$e\geq 2$$ is finite, and Vazirani showed the same for the case $$|q|=\infty$$.

My question now is the following: Has an analogous result been shown for $$q=1$$, where we have to replace $$e$$ by the characteristic of $$K$$, or rather the additive order of $$q=1$$?