Conjecture on completeness of Chalykh’s families for the harmonic chain

Prove that the families of solutions listed in Chalykh (as cited by the authors) provide a complete classification of non-terminating solution families to the multi-dimensional harmonic Darboux chain τ_n Δ τ_{n+1} − 2(∇τ_{n+1} · ∇τ_n) + τ_{n+1} Δ τ_n = 0 in all dimensions.

Background

In discussing the multi-dimensional harmonic chain of homogeneous polynomials that generate algebraically integrable Schrödinger operators, the authors highlight that full classification of non-terminating solutions in all dimensions is open and reference a conjecture regarding completeness.

They specifically note a conjecture that the families listed in Chalykh’s work constitute a complete solution of the classification problem, inviting a proof to settle this conjecture.