Tate's equivalence conjecture for BSD and Artin–Tate

Let $f\colon X\rightarrow C$ be a smooth projective surface fibration over a smooth projective curve over $k=\mathbb{F}_q$, with smooth generic fibre $X_\eta$ over $K=k(C)$, and let $J=\operatorname{Jac}(X_\eta)$. Tate's equivalence conjecture. In this situation, the Artin–Tate conjecture holds for $X$ if and only if the full Birch–Swinnerton-Dyer conjecture holds for $J$. The source later reports that Kato–Trihan proved this equivalence using the equivalences between the BSD rank statement, finiteness of a Tate–Shafarevich component, and full BSD.