Large-gap conjecture for algebraically integrable differential operators

Let $L$ be an algebraically integrable differential operator of order $n\ge 2$, with gaps $q_1,\dots,q_{n-1}$ in its local indices, and call $L$ fully cyclically symmetric when it has the stated $\mathbb Z_n$ symmetry. Large-gap conjecture. There exists $N\in \mathbb Z_+$ such that, whenever $q_j\ge N$ for every $j=1,\dots,n-1$, the only algebraically integrable operators with these gaps are fully cyclically symmetric. The conjecture is explicitly open even for $n=3$, although it has computational support and partial results; for $n=2$ it holds with $N=0$.