The non-smoothness conjecture for Lascar groups

Let $T$ be a first-order theory, and let $Gal_0(T)$ denote the kernel of the canonical map from the Lascar group $Gal_L(T)$ to the Kim--Pillay group $Gal_{KP}(T)$. Lascar-group non-smoothness conjecture. $Gal_0(T)$ is either trivial or non-smooth. This is presented as a special case of the conjecture for Lascar equivalence on KP-classes and concerns the Borel complexity of the corresponding quotient.