The nested 2-restricted Specht classification conjecture

Let $I$ be the index set of the affine type ${\tt A}_1^{(1)}$, let $\theta\in\mathbb{Z}_{\geq0}I$, and let $\mathcal{NP}_{\mathrm{2res}}(\theta)$ be the set of skew diagrams $\lambda\backslash\mu$ with $\lambda,\mu$ properly nested 2-restricted partitions of a common residue class and content $\theta$. For each such skew diagram, let $\boldsymbol{\mathsf{S}}(\lambda\backslash\mu)$ be its Specht module and $R_\theta$ the corresponding KLR algebra. Then nested 2-restricted Specht classification conjecture. For every $\lambda\backslash\mu\in\mathcal{NP}_{\mathrm{2res}}(\theta)$, $\boldsymbol{\mathsf{S}}(\lambda\backslash\mu)$ has simple head, and

$$\left\{L(\lambda\backslash\mu):=\operatorname{hd}\boldsymbol{\mathsf{S}}(\lambda\backslash\mu)\mid \lambda\backslash\mu\in\mathcal{NP}_{\mathrm{2res}}(\theta)\right\}$$

is a complete and irredundant list of simple $R_\theta$-modules up to isomorphism and grading shift. This would give an alternate classification of the simple modules for the type ${\tt A}_1^{(1)}$ KLR algebra, extending known cases involving ordinary and core-deleted skew diagrams.