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. :)