Boundary Gibbs-state conjecture for the spectral gap of PEPS parent Hamiltonians

Let a PEPS have boundary states, namely the one-dimensional mixed states associated with the PEPS by the bulk-boundary correspondence. Let its parent Hamiltonian be the corresponding bulk Hamiltonian. A one-dimensional Hamiltonian is short-range if it can be written as $H=\sum_{i,j}h_{i,j}$, where $h_{i,j}$ acts non-trivially only on spins $i$ and $j$ and

$$\|h_{i,j}\|\leq Je^{-\alpha\lvert i-j\rvert}.$$

Boundary Gibbs-state conjecture. The parent Hamiltonian of the PEPS is gapped if and only if the boundary states can be written as Gibbs states of one-dimensional short-range Hamiltonians,

$$\rho=e^{-\beta H}.$$

This conjecture relates the bulk spectral-gap problem to the effective one-dimensional boundary theory and is motivated by numerical evidence and the bulk-boundary correspondence for PEPS. The general spectral-gap problem is undecidable for PEPS, and explicit counterexamples to universal gappedness are known; the conjecture concerns the stated correspondence for boundary states.