Adelic Mordell–Lang conjecture for Brauer–Manin components

Let $k$ be a global field, let $A$ be an abelian variety over $k$, and let $X\subset A$ be a closed subvariety. Assume that $A_{\overline{k}}$ has no positive-dimensional isotrivial quotient. A coset means a translate of an abelian subvariety, and write $X(\mathbb A_k)^{\operatorname{Br}}$ for the adelic points orthogonal to the Brauer group. Adelic Mordell–Lang conjecture. There exists a finite union of cosets

$$Y=\bigcup_i C_i$$

contained in $X$ such that every connected component of

$$X(\mathbb A_k)\cap A(\mathbb A_k)^{\operatorname{Br}}$$

contains a point of $Y(\mathbb A_k)$. This was suggested for coset-free $X$ over number fields, while the stated global-field formulation remains an open problem in the supplied source.