The group quotient non-smoothness conjecture for G^{000}/G^{00}

Let $G$ be a group definable or $*$-definable over $\emptyset$ in a monster model, and let $E_G$ be the associated Borel equivalence relation coding the quotient $G^{00}/G^{000}$. Group quotient non-smoothness conjecture. If $G^{000}\ne G^{00}$, then $E_0\leq_B E_G$. By the paper's identification of $E_G$ with a corresponding Lascar-equivalence quotient, this asserts non-smoothness whenever the two connected components differ.