PEPS boundary-state locality necessity conjecture

Let $\Lambda$ be the underlying lattice, let $A\subset\Lambda$ be any rectangular region, and let $H_A$ denote the corresponding parent Hamiltonian restricted to $A$. A boundary state is the effective state associated with the boundary of such a region. PEPS boundary-state locality conjecture. If $H_A$ is gapped for any rectangular region $A\subset\Lambda$, then the boundary states of $A$ are close to a Gibbs state of a one-dimensional Hamiltonian with exponentially decaying interactions, and they are approximately factorizable.

This conjecture asks whether the boundary-state assumptions sufficient for proving a bulk gap are also necessary. It is motivated by local indistinguishability, LTQO, recovery-map results, and numerical evidence, but the source states that no proof is currently available.