Rational large-gap conjecture for algebraically integrable differential operators

Let $L$ be a rational algebraically integrable differential operator with gaps $q_1,\dots,q_{n-1}$, and call it fully cyclically symmetric when it has the stated cyclic symmetry. Rational large-gap conjecture. The large-gap conjecture holds in the rational case. Thus, for sufficiently large gaps, the only algebraically integrable rational operators are the fully cyclically symmetric ones. The source states this as a conjectural rational analogue and gives no resolution.