We recently defined invariants of contact 3-manifolds using a version of
instanton Floer homology for sutured manifolds. In this paper, we prove that if
several contact structures on a 3-manifold are induced by Stein structures on a
single 4-manifold with distinct Chern classes modulo torsion then their contact
invariants in sutured instanton homology are linearly independent. As a
corollary, we show that if a 3-manifold bounds a Stein domain that is not an
integer homology ball then its fundamental group admits a nontrivial
homomorphism to SU(2). We give several new applications of these results,
proving the existence of nontrivial and irreducible SU(2) representations for a
variety of 3-manifold groups.