Finite-dimensionality dichotomy for lattice models over background phases

Let $\mathsf{B}^{n+1}$ be a background phase and let $\mathsf{C}^{n+1}$ be an $(n+1)$D anomaly-free topological order. An MCLP lattice model over $\mathsf{B}^{n+1}$ has a local spin space $V$ and realizes $\mathsf{C}^{n+1}$. Finite-dimensionality dichotomy conjectures. If $\mathsf{C}^{n+1}$ is Morita equivalent to $\mathsf{B}^{n+1}$, then there is an $(n+1)$D MCLP lattice model over the background phase $\mathsf{B}^{n+1}$ realizing $\mathsf{C}^{n+1}$ such that $V$ is finite dimensional. If $\mathsf{C}^{n+1}$ is not Morita equivalent to $\mathsf{B}^{n+1}$, then there is an $(n+1)$D MCLP lattice model over $\mathsf{B}^{n+1}$ realizing $\mathsf{C}^{n+1}$ such that $V$ is necessarily infinite dimensional. These conjectures propose a sharp relation between Morita equivalence to the background phase and the dimensionality of the local spin space in such lattice models.