Homomorphism conjecture for fixed-point Lie algebras and quiver varieties

Let $\Gamma$ be a Dynkin diagram, let $\theta$ be the involution determined by $w_0(\alpha_i)=-\alpha_{\theta(i)}$, and let $\mathfrak g_{\Gamma}$ be the associated simple Lie algebra. Let $\sigma$ be the involution defined by $e_i\mapsto f_{a\theta(i)}$, $f_i\mapsto e_{a\theta(i)}$, and $h_i\mapsto-h_{a\theta(i)}$, and write $\mathfrak g_{\Gamma}^{\sigma}$ for its fixed-point Lie subalgebra. Let $U(\mathfrak g_{\Gamma}^{\sigma})$ be its universal enveloping algebra, and let $\mathfrak Y_{\zeta}(\mathbf w)$ denote the relevant quiver variety. Homomorphism conjecture. There is a nontrivial algebra homomorphism

$$U(\mathfrak g_{\Gamma}^{\sigma})\to \mathrm H_{top}(\mathfrak Y_{\zeta}(\mathbf w)).$$

This conjecture proposes an action of the fixed-point Lie algebra through the top homology of the quiver variety and is formulated using the coassociativity and equation~; no resolution is stated in the supplied text.