Higher-degree Hamiltonian conjecture for affine Gaudin models

Let $E_+$ be the set of positive exponents of the underlying Lie algebra, let $\varphi(z)$ be the twist function, and let $h^\vee$ be the dual Coxeter number. The quadratic Hamiltonian is denoted by $\mathscr{H}(z)$. Higher-degree Hamiltonian conjecture. For every positive exponent $d\in E_+$ with $d>1$, there exist operators $S^d(z)$ of degree $d+1$ such that, for $p,q\in E_+\setminus\{1\}$,

$$[ S^p(z), S^q(w) ] = (h^\vee\partial_z-p\varphi(z))A_{pq}(z,w)+(h^\vee\partial_w-q\varphi(w))B_{pq}(z,w),$$

and, for $p\in E_+\setminus\{1\}$,

$$[ S^p(z),\mathscr{H}(w) ]=(h^\vee\partial_z-p\varphi(z))A_{p1}(z,w),$$

for suitable operators $A_{pq}(z,w)$, $B_{pq}(z,w)$ and $A_{p1}(z,w)$. These relations would produce commuting higher-degree quantum Hamiltonians after integration against the appropriate twist-function factors; their general existence is conjectural.