Larose–Tesson characterization conjecture for logspace CSPs

Let $\mathbb{A}$ be a finite relational structure that is a core, and let $\operatorname{CSP}(\mathbb{A})$ denote the problem of deciding whether an input structure admits a homomorphism to $\mathbb{A}$. Let the algebra of polymorphisms of $\mathbb{A}$ have the indicated congruence properties. Larose–Tesson's CSP complexity characterization conjecture. The following equivalences should hold: (i) $\operatorname{CSP}(\mathbb{A})$ is solvable in nondeterministic logspace if and only if the algebra of polymorphisms of $\mathbb{A}$ is congruence join-semidistributive; (ii) $\operatorname{CSP}(\mathbb{A})$ is solvable in logspace if and only if that algebra is congruence join-semidistributive and congruence $n$-permutable for some $n$. These conjectures seek algebraic characterizations of the CSPs in NL and L; the source says that the hardness directions had been proved and presents the displayed formulation as widely discussed.