Nakajima's canonical tensor-product isomorphism conjecture for quiver varieties

Let \tilde{\frac?} be the tensor product variety associated with dominant weights $\lambda_1,\lambda_2$, and let $\omega^1,\omega^2$ be the corresponding framing data, with $\omega=\omega^1+\omega^2$. The Borel–Moore homology groups $\mathbf{H}(\mathfrak{L}(\omega^1))$, $\mathbf{H}(\mathfrak{L}(\omega^2))$, and $\mathbf{H}(\tilde{\mathfrak{Z}}(\omega))$ carry their natural $\mathfrak{g}$-module structures. Nakajima's canonical tensor-product isomorphism conjecture. There exists a unique isomorphism of $\mathfrak{g}$-modules

$$\tilde{\varphi}: \mathbf{H}(\mathfrak{L}(\omega^{1})) \otimes \mathbf{H}(\mathfrak{L}(\omega^{2})) \rightarrow \mathbf{H}(\tilde{\mathfrak{Z}}(\omega))$$

such that its restriction to $[\mathfrak{L}(0,\omega^{1})]\otimes \mathbf{H}(\mathfrak{L}(\omega^{2}))$ equals the composition of the Thom isomorphism

$$\mathbf{H}^{\mathrm{BM}}_{\mathrm{top}}(\tilde{\mathfrak{Z}}_{1},\mathbb{Q})\cong \mathbf{H}(\mathfrak{L}(\omega^{2}))$$

and the inclusion $\mathbf{H}^{\mathrm{BM}}_{\mathrm{top}}(\tilde{\mathfrak{Z}}_{1},\mathbb{Q})\rightarrow \mathbf{H}(\tilde{\mathfrak{Z}}(\omega))$. The conjecture identifies the tensor product of the two Nakajima modules canonically with the homology of the tensor product variety. It was proved for ADE quiver varieties by Nakajima; the supplied status evidence records the uniqueness property for the relevant irreducible components, while the general status is open.