we present results obtained by v2DM with the accurate three-index ($\mathcal{PQGT}'$) conditions (see Section \[intro\]) on a $6\times6$ lattice at various fillings and for different values of the on-site repulsion $U$. In Section \[intro\] we give a short introduction to the general framework behind v2DM, and show how the symmetry present in the model can be exploited to obtain a substantial computational speed up. Subsequently we present and discuss the exact variational *lower bound* results on the $6\times6$ lattice obtained using the computationally heavy $\mathcal{PQGT}'$ conditions and the much faster but less accurate two-index ($\mathcal{PQG}$) conditions. We compare these with an exact variational *upper bound* obtained through a DMRG framework developed by one of us [@sebastian]. In Section \[pure\_cluster\] we look for a way to recover three-index precision without losing computational efficiency, by imposing three-index constraints on local clusters. In Section \[extcluster\] we derive new constraints which extend these local cluster constraints in a way that reflects the open-system nature of a cluster. We demonstrate the feasibility of this approach by performing a proof-of-principle calculation imposing a subset of these constraints. It is shown that the computationally very expensive $\mathcal{PQGT}'$ results can be closely approximated by imposing only a limited 