Extend the quantum G_n tower beyond G_3 and clarify its relation to the quantum Lax pair

Construct higher-order quantum conserved operators forming a tower analogous to the classical quantities G_n=Tr[ℒ^n], beyond the sextic operator shown in equation (G3q), and determine their relationship to the quantum Lax pair (\ref{defLMq}), including whether they can be systematically derived from that Lax construction.

Background

In addition to N and M, the authors provide a quartic conserved operator H_min and recover a sextic conserved operator (G3q) inspired by the classical Tr[ℒ^3] structure. However, a general method to continue this tower is missing.

Clarifying how to extend this quantum G_n tower and how it ties into the proposed quantum Lax pair would unify the conservation-law framework and potentially yield a full hierarchy paralleling the classical integrable structure.