Classification conjecture for third-order one-pole operators

Consider algebraically integrable third-order differential operators $L$ with one pole, in the notation of the classification subsection, and let $q$ and $r$ be the associated gap parameters. The $\mathbb Z_3$-symmetric operator is characterized by $c=e=g_2=0$. Third-order classification conjecture. For $q\ge r\ge 14$, there are no algebraically integrable operators $L$ apart from the $\mathbb Z_3$-symmetric one. Consequently, all algebraically integrable third-order operators with one pole are the ones described in that subsection. This is presented as the paper’s classification conjecture, with no resolution supplied.