The generalized pushout conjecture for little-cube categories

Let ${\cal C}$ be an appropriately chosen $A_{\infty}$-operad, with ${\cal C}_1$ denoting the little $1$-cube operad, and suppose its spaces fit together freely enough. Let $\mathcal C$-categories and $\mathcal C$-groupoids be the corresponding categories and grouplike categories.

Generalized pushout conjecture. There is a notion of pushout of ${\cal C}$-categories such that the loop space functor from spaces to ${\cal C}$-groupoids takes pushouts to pushouts. The little $1$-cube operad ${\cal C}_1$ is conjectured to be an appropriate choice.

This would give a categorical pushout model for loop spaces and extend the classical generalized Seifert–van Kampen perspective. The source does not provide evidence of resolution.