There's no need to require irreducibility. If $A=E_1 \times E_2$ is a product of elliptic curves the conjecture is true, by modularity of elliptic curves combined with Yoshida's lifting from pairs of classical cusp forms to $Sp_4$. If $K$ is a real quadratic field and $E/K$ is a modular elliptic curve, then Yoshida's conjecture is true for the surface $A=\mathrm{Res}_{K/\mathbb{Q}}(E)$. This follows from a theorem of Johnson-Leung and Roberts; see arxiv 1006.5105. Presumably any individual $A$ can be done "by hand" using Faltings-Serre plus some serious computational cleverness.

Sorry if you already know all these things. :)

There's no need to require irreducibility. If $A=E_1 \times E_2$ is a product of elliptic curves the conjecture is true, by modularity of elliptic curves combined with Yoshida's lifting from pairs of classical cusp forms to Siegel modular forms of genus two. If $K$ is a real quadratic field and $E/K$ is a modular elliptic curve, then Yoshida's conjecture is true for the surface $A=\mathrm{Res}_{K/\mathbb{Q}}(E)$. This follows from a theorem of Johnson-Leung and Roberts; see arxiv 1006.5105. Presumably any individual $A$ can be done "by hand" using Faltings-Serre plus some serious computational cleverness.

Sorry if you already know all these things. :)