Kontsevich's cosheaf conjecture for Fukaya categories of Liouville sectors

A Liouville sector is a Liouville manifold with boundary satisfying the sectoriality conditions needed for its Fukaya category. The Fukaya categories of such sectors form a precosheaf under inclusions.

Kontsevich's cosheaf conjecture. Fukaya categories of Liouville sectors form a cosheaf of categories.

This is the strengthening of the precosheaf result needed in the paper's constructible-sheaf approach to mirror symmetry. It was originally conjectured by Kontsevich and, according to the source, had not yet been proved by the cited authors.