The Lascar-group upper-bound conjecture for countable tuples

Let $T$ be a theory with monster model ${\EuFrak C}$, let $\overline{a}$ be a countable tuple from ${\EuFrak C}$, and let $E_L^{\overline a}$ and $F_L^{\overline a}$ denote respectively Lascar equivalence on the type space of $\overline a$ and its restriction to the Kim--Pillay class of $\overline a$. Let $Gal_L(T)$ be the Lascar group and $Gal_0(T)$ the kernel of its canonical map to $Gal_{KP}(T)$. Lascar-group upper-bound conjecture. The Borel cardinality of $E_L^{\overline a}$ is at most that of $Gal_L(T)$, and the Borel cardinality of $F_L^{\overline a}$ is at most that of $Gal_0(T)$. The claim is motivated by placing any countable tuple inside a countable elementary substructure; its general validity is left open.