Equality of finite abelian descent, Brauer, and closure sets for smooth subvarieties of abelian varieties

Let $k$ be a global field, let $A$ be an abelian variety over $k$, and let $X\subset A$ be a smooth closed subvariety. Write $X(\mathbb{A}_k)_\bullet^{\operatorname{Br}}$ for the Brauer–Manin set, $X(\mathbb{A}_k)_\bullet^{\textup{f-ab}}$ for the finite abelian descent set, and $\overline{X(k)}$ for the topological closure of $X(k)$ in $X(\mathbb{A}_k)_\bullet$. Then the containments in the diagram

$$\overline{X(k)}\subset X(\mathbb{A}_k)_\bullet^{\operatorname{Br}}\subset X(\mathbb{A}_k)_\bullet^{\textup{f-ab}}$$

and their containments in

$$X(\mathbb{A}_k)_\bullet\cap\overline{A(k)}\subset X(\mathbb{A}_k)_\bullet\cap A(\mathbb{A}_k)_\bullet^{\operatorname{Br}}\subset X(\mathbb{A}_k)_\bullet\cap A(\mathbb{A}_k)_\bullet^{\textup{f-ab}}$$

are all equalities. This is the conjecture that finite abelian descent and Brauer–Manin conditions do not give finer sets than the closure of rational points in this setting. The question is related to conjectures and problems of Skorobogatov, Stoll, Poonen, and Voloch; the supplied source does not establish whether the conjecture is resolved.