Does the Dedekind cubical sets interpretation present spaces?

Determine whether the Quillen model structure arising from the interpretation of homotopy type theory in Dedekind cubical sets (the cartesian cube category equipped with connections) presents the homotopy theory of spaces. Concretely, ascertain whether this model category is Quillen equivalent to a standard presentation of spaces such as the Kan–Quillen model structure on simplicial sets, or, equivalently, whether its weak equivalences coincide with those in the corresponding test model structure.

Background

The paper associates Quillen model structures to constructive cubical interpretations of homotopy type theory and investigates whether they present the homotopy theory of spaces. Several existing cubical interpretations (e.g., BCH, CCHM, minimal cartesian) are known not to present spaces. The authors introduce an equivariant modification for cartesian cubical sets that does present spaces and coincides classically with the test model structure.

Among other cube categories, the Dedekind cube category (cartesian structure with connections) remains unresolved. Establishing whether its associated model structure presents spaces would clarify the landscape of constructive cubical models and their relationship to classical homotopy theory.