  Building on the theory of infinitesimal Newton--Okounkov bodies and previous
work of Lazarsfeld--Pareschi--Popa, we present a Reider-type theorem for higher
syzygies of ample line bundles on abelian surfaces. As an application of our
methods we confirm a conjecture of Gross and Popescu on abelian surfaces with a
very ample primitive polarization of type $(1,d)$, whenever $d\geq 23$.
