The natural conjecture for subgroups of direct products of limit groups

Let $\Gamma_1,\dots,\Gamma_n$ be limit groups, and let $S$ be a subgroup of $\Gamma_1\times\cdots\times\Gamma_n$ of type $FP_n(\mathbb{Q})$.

Direct-product conjecture. $S$ has a subgroup of finite index that is itself a direct product of $n$ or fewer limit groups.

This conjecture would give a positive answer to a question of Sela. The preceding results establish special cases, including subdirect products under additional hypotheses; the general statement remains open.