Finiteness conjecture for subdirect products of non-abelian limit groups

Let $L_1,\ldots,L_n$ be non-abelian limit groups with $n\geq 2$, let $D=L_1\times\cdots\times L_n$, and let $H\leq D$ be a subdirect product that intersects each factor non-trivially. Let $k\geq 2$ be an integer. For a group $G$, write $H_i(G;\mathbb{Q})$ for its $i$-th homology group with rational coefficients, and let $\operatorname{Depth}H$ denote the depth of $H$ in the direct product. Finiteness conjecture. The following conditions are equivalent: $H$ is of type $\mathrm{F}_k$; $H$ is of type $\mathrm{FP}_k(\mathbb{Q})$; $H_i(H';\mathbb{Q})$ has finite $\mathbb{Q}$-dimension for every $i\leq k$ and every finite-index subgroup $H'\leq H$; and $\operatorname{Depth}H\leq n-k$. This conjecture proposes a precise equivalence between finiteness properties, homological finiteness, and depth for subdirect products of non-abelian limit groups, extending the relationship between these invariants in direct-product settings. Its resolution is not supplied in the source.