Elliptic Lie-algebra representation conjecture via Steinberg correspondences

Let $R$ be the relevant elliptic root system, let $\delta_E$ and $\delta_{\mathrm{pt}}$ be the indicated imaginary-root classes, and let $Z_{\alpha+a\delta_E+b\delta_{\mathrm{pt}},\gamma}$ be the Steinberg-correspondence cycles acting on the equivariant cohomology of the framed moduli spaces $M^{tf}(v;\mathcal O_{D_\infty})$. Let $w^{a,b}_\gamma$ and $x^{a,b}_\alpha$ denote the generators of the elliptic Lie algebra and let $\hbar$ be the equivariant parameter. Elliptic representation conjecture. After a change of basis in the imaginary root spaces, the assignment

$$w^{a,b}_\gamma\longmapsto\bigsqcup_v Z_{a\delta_E+b\delta_{\mathrm{pt}},\gamma},$$ $$x^{a,b}_\alpha\longmapsto\bigsqcup_v Z_{\alpha+a\delta_E+b\delta_{\mathrm{pt}}},\qquad \alpha\in R\setminus\{0\},$$

defines a representation of $\mathfrak g_R\otimes\mathbb C[\hbar]$ on

$$\bigoplus_v H^*_T(M^{tf}(v;\mathcal O_{D_\infty})).$$

This would extend the real- and imaginary-root convolution representations to the full elliptic Lie algebra; the supplied text gives no resolution.