Modularity conjecture for elliptic curves over function fields

Let $X$ be a smooth, projective, geometrically irreducible curve with function field $K$, and let $E$ be an elliptic curve over $K$ with conductor $N$ and irreducible associated Galois representation. Let $\overline{\operatorname{Sht}}^2(\Gamma_0(N))$ denote the compactified moduli space of shtukas with level $\Gamma_0(N)$. Modularity conjecture. There exists a special kind of correspondence between $E\times E$ and $\overline{\operatorname{Sht}}^2(\Gamma_0(N))$. This conjectural correspondence is presented as a motivation for studying rank-two shtuka moduli spaces; no construction or resolution is supplied in the text.