Promotion conjecture for coherators with systems of inverses

Let $\mathfrak{D}$ be a coherator for $\infty$-categories, and suppose it can be endowed with a system of inverses. A coherator is contractible when it is a coherator for $\infty$-groupoids. Promotion conjecture. A coherator for $\infty$-categories is contractible provided it can be endowed with a system of inverses. This would identify Grothendieck $\infty$-groupoids with $\infty$-groupoids à la Batanin, up to essential equivalence, and would solve the relevant extension problem for the path-object construction.