The conjecture that all state-sum Hamiltonian schemas are topological

A state-sum Hamiltonian schema is obtained from a topological state-sum construction: for a cellulation $\Delta$ of a space manifold $Y$, the state-sum on $Y\times I$ defines an idempotent map $\tilde{Z}(Y\times I)$, and the Hilbert space is $V(Y)=\operatorname{Im}(\tilde{Z}(Y\times I))$. The associated local Hamiltonian terms impose admissibility and zero-flux conditions. The conjecture. All state-sum Hamiltonian schemas are topological. If true, every Hamiltonian schema arising from such a state-sum construction would satisfy the defining topological invariance properties. The supplied context explains the construction of the ground subspace and local terms but gives no resolution of this claim.