Homotopy classification conjecture for spaces of two-dimensional gapped Hamiltonians

Let $\mathcal{S}$ be a connected component of the space of two-dimensional gapped Hamiltonians realizing a fixed topological order. Let invertible states be identified under equivalence by quantum circuits and stabilization by trivial states, with tensor product as the group operation. Let an abelian anyon and a modification mean, respectively, an abelian anyon of the topological order and a modification in that theory. Homotopy classification conjecture. The fundamental group of $\mathcal{S}$ is isomorphic to the product of the group of invertible states and the group of automorphisms of the topological order. Maps from $S^2$ to $\mathcal{S}$ with a chosen basepoint are classified by a pair consisting of an invertible state in zero dimensions and an abelian anyon. Maps from $S^3$ to $\mathcal{S}$ with a chosen basepoint are classified by a modification. All higher homotopy groups are trivial modulo invertible states.

This conjecture proposes a broad homotopy-theoretic description of the space of gapped Hamiltonians, extending the relation between paths, invertible states, domain walls, and topological-order automorphisms. The source says it is supported by specific cases and general arguments, but gives no complete resolution.