Classification of one-dimensional gapped phases by boundary-state dimensions

Let $\Phi_0$ and $\Phi_1$ be two translation-invariant finite-range interactions. For $i=0,1$, suppose there are integers $l_i$ and $r_i$ such that

$${\mathcal S}^{\mathbb Z}_i\cong{\mathcal E}_1,\qquad {\mathcal S}^{(-\infty,0]}_i\cong{\mathcal E}_{r_i},\qquad {\mathcal S}^{[1,+\infty)}_i\cong{\mathcal E}_{l_i}.$$

Here ${\mathcal S}_i^\Gamma$ denotes the ground-state space on the lattice $\Gamma$, ${\mathcal E}_d$ is the state space of a $d$-dimensional quantum system, $A\cong B$ means isomorphism of convex state sets, and ${\mathcal S}_0^\Gamma\sim {\mathcal S}_1^\Gamma$ means that the two state spaces are related by a quasi-local automorphism. Boundary-state classification conjecture. The following are equivalent: (1) $r_0=r_1$ and $l_0=l_1$; (2) there is a continuously differentiable family of translation-invariant finite-range interactions $\Phi(s)$, $s\in[0,1]$, with $\Phi(0)=\Phi_0$ and $\Phi(1)=\Phi_1$, whose models have a uniform spectral gap $\gamma>0$; (3) ${\mathcal S}_0^\Gamma\sim {\mathcal S}_1^\Gamma$ for $\Gamma=\mathbb Z,(-\infty,0]$, and $[1,+\infty)$. This conjecture proposes that the dimensions of the half-chain boundary ground-state spaces completely classify these one-dimensional gapped systems, equivalently characterizing connectedness through uniformly gapped paths and quasi-local equivalence of ground states.