The m-step Darboux conjecture for exceptional orthogonal polynomial systems

Let an $\mathrm{X}_m$ orthogonal polynomial system be an exceptional orthogonal polynomial system of codimension $m$, and let a Darboux transformation be a transformation of Sturm–Liouville operators that produces a new eigenfunction system. A classical OPS denotes a classical orthogonal polynomial system.

m-step Darboux conjecture. Every $\mathrm{X}_m$ orthogonal polynomial system for any codimension $m$ can be obtained by applying a sequence of at most $m$ Darboux transformations to a classical OPS.

The statement is proved in the paper for $m\leq 2$, extending the known codimension-one result. Its extension to arbitrary codimension is stated as yet to be proved; the classification of systems obtainable by mixed-type multistep Darboux transformations also remains incomplete.