The equivalent modularity criteria for simple abelian varieties

Let $A$ be a simple abelian variety of dimension $g$ defined over $\mathbb Q$. Let ${\mathcal A}_{g,0}^{\rm tor}(N)$ be the toroidal Siegel modular variety, let $J_{g,0}(N)$ denote its Jacobian, and let $\rho_{A,\ell}$ denote the associated $\ell$-adic Galois representation.

Equivalent modularity criteria. The following statements are equivalent:

1. $A$ is modular.
2. There exists a non-constant holomorphic mapping

$${\mathcal A}_{g,0}^{\rm tor}(N)\longrightarrow A$$

for some positive integer $N$. 3. There exists a non-constant holomorphic mapping

$$J_{g,0}(N)\longrightarrow A$$

for some positive integer $N$. 4. $\rho_{A,\ell}$ is modular for any prime $\ell$.

This criterion packages several formulations of modularity for a simple abelian variety, relating modular parametrizations, the Jacobian of a Siegel modular variety, and the associated $\ell$-adic representations. The source proposes it as a conjecture and supplies no resolution status.