Geometric representation conjecture for symmetric pairs via sigma-fixed Nakajima varieties

Let $(\mathfrak g,\mathfrak k)$ be a symmetric pair listed in Table 1 in Section~. Let the relevant Nakajima varieties carry an involution $\sigma$, and let the associated Steinberg-type variety have a $\sigma$-fixed-point subvariety. Its top Borel–Moore homology is then defined, and $U(\mathfrak k)$ denotes the enveloping algebra of $\mathfrak k$. Geometric representation conjecture. There is a nontrivial algebra homomorphism

$$U(\mathfrak k)\longrightarrow H^{\mathrm{BM}}_{\mathrm{top}}(\text{$\sigma$-fixed-point Steinberg-type variety}).$$

This conjecture proposes a geometric representation-theoretic realization of the enveloping algebra of $\mathfrak k$ using $\sigma$-quiver varieties. The source indicates supporting evidence in Proposition~ and equation~, but does not state that the conjecture has been proved.