Subgroups of direct products of limit groups

If $G_1,\mathrm{.}\mathrm{.}\mathrm{.},G_n$ are limit groups and $S\subset G_1\times \mathrm{.}\mathrm{.}\mathrm{.}\times G_n$ is of type ${\text{FP}}_n(\mathbb{Q})$ then $S$ contains a subgroup of finite index that is itself a direct product of at most $n$ limit groups. This settles a question of Sela.