The upper-ledge Specht-module conjecture

Let $I$ be the index set of the affine type ${\tt A}_1^{(1)}$, let $\theta\in\mathbb{Z}_{\geq0}I$, and let $\mathsf{D}\in\mathcal{UL}(\theta)$ be an upper ledge diagram. Let $\xi(\mathsf{D})$ be the associated skew diagram, $\boldsymbol{\mathsf{S}}(\xi(\mathsf{D}))$ its Specht module, and $L(\mathsf{D})$ the simple module indexed by $\mathsf{D}$. Then upper-ledge Specht-module conjecture. The module $\boldsymbol{\mathsf{S}}(\xi(\mathsf{D}))$ has simple head isomorphic to $L(\mathsf{D})$ up to a grading shift. The conjecture proposes a Specht-module realization of the simple modules indexed by the intermediary upper-ledge crystal; the source presents it as part of a conjectural alternate classification of KLR-algebra representations.