Maximality conjecture for the noncommutative spectral-curve Hamiltonians

Let $A$ be the free associative algebra of the system, let $[A,A]$ be its commutator subspace, and write the coefficients of the noncommutative spectral curve as

$$\operatorname{Tr}\,L(\lambda)^k=\sum_{j=-k}^k H_{k,j}\lambda^j.$$

The induced Lie bracket on $A/[A,A]$ is the one associated with the system. Spectral-curve maximality conjecture. The image of $\operatorname{Span}(H_{k,l})$ in $A/[A,A]$ under the natural projection is a maximal commutative Lie subalgebra of $A/[A,A]$ with respect to this induced Lie bracket. The source reports computations showing commutativity for $k,m\leq 5$ and arbitrary indices, but maximality remains conjectural.