The totally symmetric polymorphism conjecture for omega-categorical CSPs

Let $\Gamma$ be an $\omega$-categorical structure with finite relational signature. A family of totally symmetric polymorphisms consists of polymorphisms of every arity whose value is invariant under every permutation of the arguments. The totally symmetric polymorphism conjecture. If $\Gamma$ has totally symmetric polymorphisms of all arities, then

$$\operatorname{CSP}(\Gamma)$$

can be solved in polynomial time. This is proposed as a generalization beyond the sub-exponential case, where the paper describes a classification and an efficient sampling algorithm; the conjecture remains open in the source.