Larose–Tesson conjecture on Kearnes–Kiss chains and NL

Let $\mathbb{H}$ be a finite structure with finite relational signature, and let its polymorphisms be the edge-preserving operations on its domain. A Kearnes–Kiss chain is the chain defined in Section~. Larose–Tesson's conjecture. If the polymorphisms of $\mathbb{H}$ contain a Kearnes–Kiss chain, then

$$\operatorname{CSP}(\mathbb{H})\text{ is in NL}.$$

The converse hardness results show that failure of the condition implies hardness for classes not believed to lie in NL. The conjecture is wide open and is regarded as one of the most difficult open problems in finite-domain constraint satisfaction.