Top-degree cocenter basis conjecture for cyclotomic quiver Hecke algebras

Let $\Lambda\in P^+$ and $\alpha\in Q^+$. Let $\mathscr{R}_{\alpha}^{\Lambda}$ be the cyclotomic quiver Hecke algebra over a ground field $K$, let $d_{\Lambda,\alpha}$ denote the relevant top degree, and let $S(\nu)$ be the element associated to a piecewise dominant sequence $\nu\in I^\alpha$. Write $\operatorname{Tr}(\mathscr{R}_{\alpha}^{\Lambda})$ for the cocenter. Top-degree cocenter basis conjecture. One has

$$\operatorname{Tr}(\mathscr{R}_{\alpha}^{\Lambda})_{d_{\Lambda,\alpha}}=K\bigl(S(\nu)+[\mathscr{R}_{\alpha}^{\Lambda},\mathscr{R}_{\alpha}^{\Lambda}]\bigr).$$

In particular, $\dim\operatorname{Tr}(\mathscr{R}_{\alpha}^{\Lambda})_{d_{\Lambda,\alpha}}=1$. This conjecture refines the indecomposability conjecture by describing the relevant top-degree cocenter component; the supplied text does not state a resolution.