The conditional Brauer–Manin conjecture for Kummer surfaces

Let $k$ be a number field and let $X$ be a Kummer surface over $k$. A 2-covering associated to $X$ means one of the 2-coverings of the abelian surfaces associated to $X$. Kummer-surface Brauer–Manin conjecture. If the 2-primary part of the Brauer–Manin obstruction to the Hasse principle is the only obstruction for all 2-coverings associated to $X$, then the same is true for $X$. The source explains that successful applications of Swinnerton-Dyer's method establish cases of this conjecture unconditionally, while the general assertion remains open.