Calogero–Moser critical-point conjecture for symmetric elliptic operators

Let $\ell\in\{3,4,6\}$, and let $L$ be an order-$\ell$ operator invariant under the group $\mathbb Z_\ell$ on an elliptic curve with the corresponding symmetry. Assume that its poles and local indices are as specified in the source: poles at the fixed points $\eta_j$ and at distinct $\mathbb Z_\ell$-orbits of points $z_1,\dots,z_N$, with the prescribed indices at the fixed and non-fixed poles. Elliptic Calogero–Moser critical-point conjecture. Algebraically integrable operators $L$ as above correspond to critical points of the lowest-degree, namely degree $\ell$, Hamiltonian of the classical crystallographic elliptic Calogero–Moser system for $\mathbb Z_\ell$, with appropriate parameters. This conjecture identifies algebraic integrability with critical points of the corresponding elliptic Calogero–Moser Hamiltonian; the source does not report a resolution.