Grothendieck's conjecture on models of weak infinity-groupoids

A globular weak $\infty$-groupoid model is any of the unspecified models of globular weak $\infty$-groupoids considered in the conjecture. Let the category of such models be equipped with a Quillen model structure, and let the category of spaces carry its usual Quillen model structure, whose weak equivalences are given by homotopy groups and whose fibrations are Serre fibrations. Grothendieck's conjecture. The category of some models of globular weak $\infty$-groupoids is equipped with a Quillen model structure that is Quillen equivalent to the category of spaces with its usual Quillen model structure. The conjecture proposes an algebraic model for homotopy types of spaces; the source presents it as motivation for cubical weak $\infty$-groupoid models and does not state a resolution.