Stability conjecture for semisimplicity of twisted cyclotomic quiver Hecke algebras

Assume $p\nmid(e+1)$, let $I=\\{0,1,\ldots,e\\}$, and let $R^{\Lambda}_n(I)$ denote the cyclotomic quiver Hecke algebra associated with a dominant integral weight $\Lambda\in P^+$. Stability conjecture. The following equivalences should hold:

1. If $\langle h_0,\Lambda\rangle\in 2\mathbb{N}$, then $R^{\Lambda}_n(I)$ is semisimple if and only if $R^{\Lambda+\Lambda_0}_n(I)$ is semisimple.
2. If $\langle h_e,\Lambda\rangle\in 2\mathbb{N}$, then $R^{\Lambda}_n(I)$ is semisimple if and only if $R^{\Lambda+\Lambda_e}_n(I)$ is semisimple.

This is motivated by the preceding semisimplicity equivalence and is proposed after the known “if” direction of the related criterion. The equivalences for arbitrary dominant integral weights remain conjectural in the supplied text.