The non-smoothness conjecture for Lascar equivalence on KP-classes

Let $X$ be an $E_{KP}$-class. The relations $E_L$ and $E_0$ are defined on spaces of types, where $E_L\!\upharpoonright\!X$ denotes the restriction of Lascar equivalence to $X$, and $E_0$ is eventual equality on $2^{\mathbb N}$. Non-smoothness conjecture. $E_L \! \upharpoonright \! X$ is either trivial or non-smooth; in the latter case, $E_0$ is Borel reducible to $E_L \! \upharpoonright \! X$. This strengthens Newelski's theorem, which says that a nontrivial restriction has continuum many classes; the conjecture asks that every such restriction be non-smooth.