Conjecture on the cosocle construction for simple cyclotomic KLR modules

Let $A\in\operatorname{Rep}({}^{\Lambda_i})$ be a simple $R(\nu)$-module with $|\nu|\geq 1$, where $i$ is not a forbidden element of $I$. Let $p$ be a cyclotomic path and let $k=r(A)$ be minimal such that $\mathcal{R}(A)={\widetilde{e}^\vee_{p;k}}A$ is a simple $R(\nu-\gamma_{p;k})$-module in $\operatorname{Rep}({}^{\Lambda_{p(-1)}})$; write $T_{p;k}$ for the corresponding module constructed from the path.

Cosocle construction conjecture. Under the same hypotheses as the existence theorem,

$$A\cong\operatorname{cosoc}\operatorname{pr}_{\Lambda_i}\operatorname{Ind}(T_{p;k})\boxtimes\mathcal{R}(A).$$

This conjecture strengthens the existence theorem's surjection by asserting that the simple module $A$ is precisely the cosocle of the induced and projected tensor product. It concerns the proposed categorification of the tensor product of the Kirillov–Reshetikhin crystal $B^{1,1}$ with a fundamental crystal; no resolution is given in the supplied text.