Frank Wagner's Baldwin–Saxl conjecture for type definable groups

Let $T$ be a dependent theory. A group $G$ is type definable if it is defined by a possibly infinite collection of formulas. Let $p(x,y)$ be a type, and let $\langle a_i\mid i<\omega\rangle$ be an indiscernible sequence such that $G_i=p(x,a_i)\leq G$.

Frank Wagner's conjecture. There is some $n$ such that, for every finite set $v\subseteq\omega$, the intersection

$$\bigcap_{i\in v}G_i$$

is equal to a sub-intersection involving $n$ of the groups $G_i$.

This is the proposed extension of the Baldwin–Saxl chain condition from definable groups to type definable groups in dependent theories. The statement is presented as the main motivation for the open question of whether infinite type definable fields in dependent theories are Artin–Schreier closed; its resolution status is not specified in the source.