The paramodular conjecture for abelian surfaces

Let $\mathcal{A}/\mathbb{Q}$ be an abelian surface of conductor $N$ with $\operatorname{End}_{\mathbb{Q}}\mathcal{A}=\mathbb{Z}$. Let $F$ be a weight $2$ cuspidal paramodular newform of level $N$ that is not a Gritsenko lift and has rational Hecke eigenvalues. The paramodular conjecture. There is a one-to-one correspondence between isogeny classes of such abelian surfaces and such forms $F$, up to scalar multiplication. Furthermore, the Hasse–Weil $L$-function of $\mathcal{A}$ equals the spinor $L$-function of $F$. This conjecture is the anticipated extension of the modularity theorem from elliptic curves to abelian surfaces; the supplied text does not state a resolution.