Modularity of higher dimensional abelian varieties

There has been recent work in some concrete cases. There's a paper by Poor and Yuen that gives computational evidence for a special case of the so-called Paramodular Conjecture. This Conjecture is described as "a precise and testable modularity conjecture for rational abelian surfaces $\mathcal{A}$ with trivial endomorphisms, $End_\mathbb{Q} \mathcal{A} = \mathbb{Z}$ in the abstract of a paper by Brumer and Kramer. To the best of my knowledge, this is the most precise version of a general prediction made by Yoshida as described in

H. Yoshida, On generalization of the Shimura-Taniyama conjecture I and II, Siegel Modular Forms and Abelian Varieties, Proceedings of the 4-th Spring Conference on Modular Forms and Related Topics, 2007, pp. 1-26.