Jetchev–Stein modularity conjecture for classes in Tate–Shafarevich groups

Let $E/\mathbf{Q}$ be an elliptic curve of conductor $N$. A $J_0$-modular abelian variety is an abelian variety over $\mathbf{Q}$ that is a quotient of $J_0(M)=\operatorname{Jac}(X_0(M))$ for some $M\geq1$. A class $c\in\Sh(E/\mathbf{Q})$ is modular if there is an embedding $\iota:E\to A$ into a $J_0$-modular abelian variety $A$ such that $c$ is visible in $A$.

Jetchev–Stein conjecture. Every class in $\Sh(E/\mathbf{Q})$ is modular.

This is the paper’s precise visibility formulation of the Jetchev–Stein conjecture. The cited work gives evidence, including modularity of classes of order $2$ or $3$ and visibility results for classes splitting over abelian extensions.