FDQC equivalence conjecture for commuting projector Hamiltonians

Let $H$ and $H'$ be local commuting projector Hamiltonians with no long-range bulk order. A Hamiltonian is FDQC equivalent to another when their ground states become equivalent after adding ancillary qudits and applying a finite-depth quantum circuit. Suppose each Hamiltonian has a gapped boundary with surface topological orders described by modular tensor categories, and let Witt equivalence refer to equivalence in the categorical Witt group.

FDQC equivalence conjecture. The Hamiltonians $H$ and $H'$ are FDQC equivalent if and only if their surface topological orders in the presence of a gapped boundary are Witt equivalent.

The conjecture identifies the Witt class of the surface order as the obstruction to finite-depth-circuit equivalence of bulk-trivial commuting projector Hamiltonians. The paper motivates both directions using boundary stacking and compactification arguments, but does not prove the equivalence in general.