Geometric categorical action conjecture for simply laced Kac–Moody algebras

Let $\mathfrak{g}$ be a Kac–Moody Lie algebra whose Dynkin diagram is simply laced and contains no loops. A geometric categorical $\mathfrak{g}$ action is the categorical action on derived categories associated with the geometry of the corresponding quiver varieties. A strong 2-representation is a 2-representation satisfying the specified KLR-algebra relations. Geometric categorical action conjecture. A geometric categorical $\mathfrak{g}$ action induces a strong 2-representation of $\mathfrak{g}$. This would extend the known geometric categorical $\mathfrak{sl}_2$ construction to Nakajima quiver varieties of simply laced loopless type; the source says that the key missing step is constructing the KLR-algebra action on the functors $\mathsf{E}$, and gives no resolution for this general statement.