The expansion conjecture for trivial Lascar group

Let $M$ be an $\omega$-categorical structure. Its Lascar group is the quotient of $\operatorname{Aut}(M)$ by the subgroup of Lascar strong automorphisms. Expansion conjecture. Any $\omega$-categorical structure has an $\omega$-categorical expansion with trivial Lascar group. The conjecture would imply that a nontrivial Lascar group cannot be an additional obstruction to finite topological rank beyond compact quotients. Its status is not resolved in the supplied text.