The level-$(0,-1)$ geometric action conjecture for affine type A

Fix $k\in I$, let $\boldsymbol v=(v_0,\ldots,v_{N-1})$ and let $\mathcal M(\boldsymbol v,\boldsymbol w)$ be the quiver variety of type $A^{(1)}_{N-1}$ with $\boldsymbol w=(\delta_{i,k})$. Let $\mathrm E_\mathrm T(\mathcal M(\boldsymbol v,\boldsymbol w))$ be its equivariant elliptic cohomology, let $\mathcal F^{(0,-1)(k)}_u$ be the stated level-$(0,-1)$ representation of $U_{t_1,t_2,p}(\mathfrak{gl}_{N,tor})$, and let $|\lambda\rangle^{(k)}_u$ and $[\lambda]$ denote respectively its partition-labelled basis vector and the corresponding $\mathrm T$-fixed-point class.

Level-$(0,-1)$ geometric action conjecture. The representation $\mathcal F^{(0,-1)(k)}_u$ of $U_{t_1,t_2,p}(\mathfrak{gl}_{N,tor})$ is equivalent to an expected level-$(0,-1)$ geometric action of the same algebra on

$$\bigoplus_{\boldsymbol v}\mathrm E_\mathrm T(\mathcal M(\boldsymbol v,(\delta_{i,k}))),$$

under the identification of $|\lambda\rangle^{(k)}_u$ with the $\mathrm T$-fixed-point class $[\lambda]$.

This conjecture asserts that the representation constructed in the paper matches the expected elliptic-cohomological action on the affine type-A quiver varieties. The source gives no resolution.