Bodirsky–Schneider–Thom conjecture on singular mixed identities in oligomorphic groups

Let $G\curvearrowright\Omega$ be a permutation group, where $\Omega$ is a countably infinite set. The action is oligomorphic if $G$ has finitely many orbits on $\Omega^n$ under its diagonal action

$$g\cdot(a_1,\dots,a_n)=(ga_1,\dots,ga_n).$$

Bodirsky–Schneider–Thom conjecture. If $G\curvearrowright\Omega$ is oligomorphic, then all mixed identities of $G$ are singular.

This conjecture predicts that oligomorphic groups, a broad class including automorphism groups of countable $\omega$-categorical structures, have no non-singular mixed identities. It was proposed by Bodirsky, Schneider, and Thom; the supplied source gives no evidence that it has been resolved.