Tate--Shafarevich group conjecture for Type i KSS Jacobians

Let $J_i\to\mathbb{P}^2$ be the common relative Jacobian of the Type $i$ KSS pair. The Tate--Shafarevich group is the group of torsors of the Jacobian fibration that are locally trivial. Tate--Shafarevich group conjecture. The Tate--Shafarevich group for $J_i/\mathbb{P}^2$ is isomorphic to $\mathbb{Z}_5$. The abstract states that the authors prove this assertion for each pair, so the conjecture is solved.