The Paramodular Conjecture for rational abelian surfaces

Let $p$ be a prime. A paramodular form here is a Hecke eigenform $F\in S^2(\Gamma^{\text{para}}[p])$ with rational eigenvalues; exclude Gritsenko lifts. Let $\mathcal{A}$ be an isogeny class of rational abelian surfaces of conductor $p$, and let $L(F,s)$ and $L(\mathcal{A},s,$Hasse--Weil$)$ denote their respective $L$-series. Paramodular Conjecture. There is a bijection between lines of such eigenforms $F$ and isogeny classes of such abelian surfaces $\mathcal{A}$, under which

$$L(\mathcal{A},s,\text{Hasse--Weil})=L(F,s).$$

The conjecture gives a correspondence between weight-two paramodular forms and abelian surfaces of prime conductor; the source also notes that analytic continuation and functional equations for the two $L$-series are conjectural in this setting.