We have something between the two in HoTT - universes of types is stratified by homotopy levels, corresponding to how many dimensions of structure a type has. A space with only points is thus a 0-type, a space with at most 1 point is a -1-type, and a space with only one is a -2-type.
The catch is that univalence is inconsistent with LEM at h-levels greater than -1, but assuming it is perfectly consistent for -1 types, which can be thought of as the "at most true" propositions of classical logic.
A lot of work was going into the cubical model, but IIRC they realized it was a dead end about a month and a half ago. Right now the most promising work looks to be formalizing the set theoretic model in NuPRL.
The catch is that univalence is inconsistent with LEM at h-levels greater than -1, but assuming it is perfectly consistent for -1 types, which can be thought of as the "at most true" propositions of classical logic.