The affine type A Kleshchev skew-Specht classification conjecture

Let $e\geq2$, let $i\in\mathbb{Z}_e$, and let $R_\theta({\tt A}_{e-1}^{(1)})$ be the KLR algebra for a root-lattice element $\theta$. Let $\mathcal{MP}_{\mathrm K}^{(i,i)}(\theta)$ be the Kleshchev ${\tt A}_{e-1}^{(1)}$-bipartitions of bicharge $(i,i)$ with content $\theta$. Then affine type A Kleshchev skew-Specht classification conjecture. For every $(\lambda^{(1)},\lambda^{(2)})\in\mathcal{MP}_{\mathrm K}^{(i,i)}(\theta)$, the Specht module $\boldsymbol{\mathsf{S}}(\lambda^{(1)}\backslash\lambda^{(2)})$ has simple head, and

$$\left\{\operatorname{hd}\boldsymbol{\mathsf{S}}(\lambda^{(1)}\backslash\lambda^{(2)})\mid(\lambda^{(1)},\lambda^{(2)})\in\mathcal{MP}_{\mathrm K}^{(i,i)}(\theta)\right\}$$

is a complete, though redundant, list of simple $R_\theta({\tt A}_{e-1}^{(1)})$-modules up to isomorphism and grading shift. This conjecture seeks a uniform skew-Specht classification in all affine type A ranks, motivated by small-rank computations.