May's generalized Seifert–van Kampen conjecture for $A_{\infty}$-groupoids

Let ${\cal C}$ be an $A_{\infty}$-operad. Associate to each space $X$ its loop space, regarded as a grouplike $\widehat{{\cal C}}$-space, and let a grouplike $\widehat{{\cal C}}$-category be called a $\widehat{{\cal C}}$-groupoid.

May's generalized Seifert–van Kampen conjecture. The loop space functor from spaces to grouplike $\widehat{{\cal C}}$-spaces takes pushouts of connected spaces to pushouts of $\widehat{{\cal C}}$-spaces, and more generally takes pushouts of not necessarily connected spaces to pushouts of $\widehat{{\cal C}}$-groupoids.

This is intended as a homotopical generalization of the Seifert–van Kampen theorem, expressing loop spaces and higher groupoids through pushout constructions. The source does not provide evidence of resolution.