Darboux classification conjecture for exceptional orthogonal polynomial systems

Let an exceptional orthogonal polynomial system have codimension $m$, and let a classical orthogonal polynomial system be given. A Darboux transformation is a transformation relating the corresponding spectral problems.

Gómez-Ullate–Kamran–Milson conjecture. Every exceptional orthogonal polynomial system of codimension $m$ can be obtained by applying a sequence of at most $m$ Darboux transformations to a classical orthogonal polynomial system.

This conjecture proposes a complete Darboux-based classification of exceptional orthogonal polynomial systems and is presented as a fundamental step toward their classification. The supplied text gives no resolution, so its status is open.