Cosocle decomposition conjecture for simple modules over cyclotomic quiver Hecke algebras

Let $A$ be a simple $R(\nu)$-module in $\operatorname{Rep}^{\Lambda_i}$ with $|\nu|\geq 1$. Let $r(A)$ and $\mathcal{R}(A)$ be as in the preceding theorem for trivial modules, and let $c(A)$ and $\mathcal{C}(A)$ be defined by the corresponding sign-module theorem. Write $T_{i;r(A)}$ and $S_{i;c(A)}$ for the associated trivial and sign modules. Cosocle decomposition conjecture. With these hypotheses and definitions,

$$A=\operatorname{cosoc}\,\operatorname{pr}_{\Lambda_i}\operatorname{Ind}T_{i;r(A)}\boxtimes\mathcal{R}(A),$$

and

$$A=\operatorname{cosoc}\,\operatorname{pr}_{\Lambda_i}\operatorname{Ind}S_{i;c(A)}\boxtimes\mathcal{C}(A).$$

The surrounding results establish surjections from the displayed induced modules onto $A$; the assertion strengthens these to identifications of $A$ with their cosocles. The supplied text gives no evidence that this claim has been proved or refuted elsewhere.