The cubical–complicial Quillen equivalence conjecture

Let $\mathsf{cSet}^+$ denote the category of marked cubical sets and let $\mathsf{PreComp}$ denote the category of marked simplicial sets (pre-complicial sets). Let $T\colon \mathsf{cSet}^+\to\mathsf{PreComp}$ be the functor introduced above. The comical and complicial model structures may each be taken in their ordinary, saturated, or $n$-trivial forms, with the same choice made on both sides. Cubical–complicial Quillen equivalence conjecture. The functor $T$ is a Quillen equivalence with respect to the (respectively saturated or $n$-trivial) comical model structure and the (respectively saturated or $n$-trivial) complicial model structure. This would identify the cubical and simplicial models for $(\infty,n)$-categories, including their saturated and $n$-trivial variants. The preceding result establishes that $T$ is a Quillen adjunction for the ordinary and $n$-trivial structures; the conjectural Quillen equivalence, particularly the saturated version, remains to be proved.