Countability conjecture for minor equivalence classes of finite-domain clones

A clone is a set of finitary operations on a domain that contains all projections and is closed under composition. Two clones are minor equivalent when each admits a minor-preserving map to the other.

Countability conjecture. The minor equivalence relation on clones on finite domains has only countably many classes.

The paper establishes this countability phenomenon for self-dual clones up to minor equivalence, while the conjecture asks for it across all clones on finite domains. An indicated first step is to prove countability for clones on a three-element set up to minor equivalence.