Egri–Larose–Tesson conjecture on Noname chains and L

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

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

As with the NL conjecture, failure of the condition yields hardness for complexity classes not believed to lie in L. The conjecture remains open, including for finite digraphs and orientations of finite trees.