The KLR character conjecture for rigid quiver representations

Let $(Q,\mathbf{d})$ be a valued quiver, let $\Omega$ be the quantum shuffle character, and let $[V]^*$ denote the dual Hall--Ringel algebra element associated with a representation $V$ of $(Q,\mathbf{d})$. KLR character conjecture. For any rigid representation $V$ of $(Q,\mathbf{d})$, the quantum shuffle character $\Omega([V]^*)$ produces the character of an irreducible KLR (quiver Hecke) representation. This conjecture would relate rigid representations of acyclic valued quivers to irreducible representations of KLR algebras and would establish the proposed KLR conjecture for acyclic skew-symmetrizable quantum cluster algebras.