DELL eigenfunction conjecture for affine Laumon-space elliptic genera

Let $N-1$ degrees of freedom of the quantum double elliptic (DELL) system be described by the current $\mathcal{O}(z)$ and let $\mathcal{Z}^{DELL}_{\text{inst}}(p,x_1,\dots, x_N)$ be an equivariant elliptic genus of the affine Laumon space. Let $\mathcal{O}_0$ denote the zeroth Fourier mode of the current, and let $\lambda(z,\textbf{a},w,p)$ be a function of $z$, $\textbf{a}$, $w$, and $p$. DELL eigenfunction conjecture. There exists a function $\lambda(z,\textbf{a},w,p)$ such that

$$\mathcal{O}(z)\mathcal{Z}^{DELL}_{\text{inst}}(p,x_1,\dots, x_N)=\lambda(z,\textbf{a},w,p)\mathcal{O}_{0}\mathcal{Z}^{DELL}_{\text{inst}}(p,x_1,\dots, x_N).$$

Equivalently, after expanding in $z$, the DELL Hamiltonians have $\mathcal{Z}^{DELL}_{\text{inst}}$ as a common eigenfunction with eigenvalues $\lambda_n(\textbf{a},w,p)$. This gives a localization construction of the formal spectrum of the quantum DELL system, whose Hamiltonian commutativity has only been checked to several orders in the elliptic parameters $p$ and $w$.