The finiteness-property conjecture for full subdirect products of curve groups

Let $G$ be a subgroup of a direct product $D$ of smooth curve groups, and let $\phi$ be the associated map to the product of the quotient groups, as in the surrounding construction. Assume that $G$ is a full subdirect product of $D$, and let $k\geq 1$. Finiteness-property conjecture. The group $G$ is of type $FP_k(\mathbb{Q})$ if and only if either $\phi$ is not injective or $G$ has a finite-index subgroup which is a direct product of at most $k$ smooth curve groups. The surrounding results establish related implications for injective maps and infinite-index subgroups, but the supplied text does not state a resolution of this equivalence.