Kleshchevization's Specht-module conjecture

Let $\boldsymbol{\kappa}=(\kappa_1,\kappa_2)$ be a bicharge, let $\boldsymbol{\lambda}\in\mathcal{MP}^{\boldsymbol{\kappa}}$, and let $\boldsymbol{\lambda}_{\mathrm K}$ denote its Kleshchevization. Write $S^{\boldsymbol{\kappa}}(\boldsymbol{\lambda})$ for the corresponding Specht module and $D^{\boldsymbol{\mu}}$ for the simple module indexed by a Kleshchev multipartition. Then Kleshchevization's Specht-module conjecture.

1. There exists a nonzero homomorphism

$$S^{\boldsymbol{\kappa}}(\boldsymbol{\lambda}_{\mathrm K})\rightarrow S^{\boldsymbol{\kappa}}(\boldsymbol{\lambda}).$$

2. $[S^{\boldsymbol{\kappa}}(\boldsymbol{\lambda}):D^{\boldsymbol{\lambda}_{\mathrm K}}]=1$.
3. If $[S^{\boldsymbol{\kappa}}(\boldsymbol{\lambda}):D^{\boldsymbol{\mu}}]>0$ for $\boldsymbol{\mu}\in\mathcal{MP}_{\mathrm K}^{\boldsymbol{\kappa}}$, then $\boldsymbol{\mu}\trianglelefteq\boldsymbol{\lambda}_{\mathrm K}$. This predicts that Kleshchevization identifies a distinguished simple constituent and controls all other Kleshchev-labelled constituents in the dominance order.