Prove simplicity of the modules associated with belts

Prove that, for every belt \Gamma, the explicitly constructed degenerate affine Hecke algebra module M_\Gamma is simple.

Background

For belt blocks, the paper constructs a module M_\Gamma for the degenerate affine Hecke algebra \mathcal{H}_pk associated with each belt \Gamma. These modules have formal characters matching the combinatorial objects used to approximate the decomposition matrix. Establishing their simplicity would support the interpretation of belts as a family of simple modules, but the paper does not prove this assertion and states it as a conjecture.

References

It seems natural that the set of belts gives a family of simple \calh_pk-modules. Despite this, we will not focus on this question in the present paper. We write our suspect as a conjecture such that it can inspire further research.

Let $\Ga$ be a belt. Then the $\calh{k}_p$-module $M_{\Ga}$ is simple.

— Ribbon blocks for centraliser algebras of symmetric groups  (2502.13867 - Fayers et al., 19 Feb 2025) in Section 4, "Belt blocks"