Subgroups of direct products of elementarily free groups

The structure of groups having the same elementary theory as free groups is now known: they and their finitely generated subgroups form a prescribed subclass E of the hyperbolic limit groups. We prove that if G(1),..., G(n) are in E then a subgroup Gamma subset of G(1) x... x G(n) is of type FPn if and only if Gamma is itself, up to finite index, the direct product of at most n groups from E. This provides a partial answer to a question of Sela.