Higher modularity conjecture for nonisotrivial elliptic curves

Let $K/\mathbf{F}_q$ be a function field and let $E/K$ be a nonisotrivial elliptic curve. For $r\geq 1$, $E$ is called $r$-modular when it is $(r,\Sigma_\infty)$-modular for every admissible subset $\Sigma_\infty$ of places in the conductor, as defined by the existence of a degree $0$ algebraic correspondence inducing surjective maps on the relevant cohomology for all geometric points. Higher modularity conjecture. A nonisotrivial elliptic curve $E/K$ is $r$-modular for all $r\geq 1$. This conjecture predicts algebraic correspondences realizing the cohomological modularity of elliptic curves over function fields; the supplied text gives no resolution status.