The effective categorification conjecture for $A_{\infty}$-operads

Let ${\cal C}$ be an $A_{\infty}$-operad, and let $\widehat{{\cal C}}$-precats and $\widehat{{\cal C}}$-categories denote the associated precategories and categories. For a $\widehat{{\cal C}}$-precat $A$, write $A_{1/}$ for its space of composable pairs of objects and morphisms.

Effective categorification conjecture. For any $A_{\infty}$-operad ${\cal C}$, there is an operation $Cat _{{\cal C}}$ from $\widehat{{\cal C}}$-precats to $\widehat{{\cal C}}$-categories such that $Cat _{{\cal C}}$ is a monad on the category of $\widehat{{\cal C}}$-precats. If $A$ is a $\widehat{{\cal C}}$-category, then the morphism $A \rightarrow Cat _{{\cal C}}(A)$ is an equivalence of $\widehat{{\cal C}}$-categories. If $A_{1/}$ is connected, calculation of $Cat _{{\cal C}}(A)$ is effective.

This conjecture proposes a constructive categorification procedure that leaves already-complete $\widehat{{\cal C}}$-categories unchanged up to equivalence. The source does not provide evidence of resolution.