Maximal product-overlap conjecture for general Haar random tensor networks

Prove that for a general Haar random tensor network, the maximal product-state overlap of the tripartite boundary state satisfies $\log \Lambda(\Psi_{A:B:C})=-\operatorname{Area}(\Gamma_{\min})+O(1)$, with the leading-order maximum achieved by the product state obtained by fixing the bond indices across the minimal bulk tripartition surface.

Background

The paper reduces the lower-bound problem for the relative entropy of entanglement to controlling the maximal overlap between the Haar random tensor-network state and arbitrary product states across the boundary tripartition $A:B:C$.

A product state constructed from the minimal tripartition surface is shown to attain overlap with the scaling expected from the surface area. The unresolved issue is whether any arbitrary product state can achieve a parametrically larger overlap. The paper proves the conjecture for networks of one, two, and three independent Haar-random tensors, but leaves it unresolved for general tensor networks.