Elliptic Ruijsenaars-Schneider eigenfunction conjecture

Let $\mathcal{H}^{e\text{RS}}_k$ be the elliptic Ruijsenaars-Schneider Hamiltonians for $k=1,\dots,N-1$, and let $\mathcal{Z}^{\text{RS}}(\textbf{a},\textbf{x})$ be a function of the equivariant parameters $\textbf{a}$ and coordinates $\textbf{x}$. Let $\mathcal{L}^{\text{aff}}_{\textbf{d}}$ be the affine Laumon space, and let $\mathfrak{q}=(\mathfrak{q}_1,\dots,\mathfrak{q}_n)$ be a string of $\mathbb{C}^\times$-valued coordinates on its maximal torus. Elliptic Ruijsenaars-Schneider eigenfunction conjecture. The eigenfunctions of the elliptic Ruijsenaars-Schneider Hamiltonians satisfy

$$\mathcal{H}^{e\text{RS}}_k\mathcal{Z}^{\text{RS}}(\textbf{a},\textbf{x})=\lambda_k(\textbf{a})\mathcal{Z}^{\text{RS}}(\textbf{a},\textbf{x}),\qquad k=1,\dots,N-1,$$

where

$$\mathcal{Z}^{\text{RS}}=\sum_{\textbf{d}}\mathfrak{q}^{\textbf{d}}\int\limits_{\mathcal{L}_{\textbf{d}}}1.$$

This identifies the eigenfunctions with the K-theoretic holomorphic equivariant Euler characteristic of the affine Laumon space and forms part of the progress on the spectrum of the elliptic Ruijsenaars-Schneider model; the supplied passage does not state a resolution.