Precompactness conjecture for quantum Riemannian 1-spaces

Fix real numbers $C$, $a$, and $b$. Consider quantum Riemannian $1$-spaces with measure whose associated Hilbert space comes from a unital involutive algebra and a state, with non-negative Ricci curvature, spectral gap at least $C$, and corresponding spectral triples having dimensional spectrum contained in $[a,b]$. Precompactness conjecture. The space of isomorphism classes of these quantum Riemannian $1$-spaces with measure is precompact in the topology defined in the source. This is a noncommutative analogue of geometric precompactness under curvature, spectral-gap, and dimension-spectrum bounds; the source gives no resolution.