A proof of Kontsevich-Soibelman conjecture

It is well known that "Fukaya category" is in fact an $A_{\infty}$-pre-category in sense of Kontsevich and Soibelman \cite{KS}. The reason is that in general the morphism spaces are defined only for transversal pairs of Lagrangians, and higher products are defined only for transversal sequences of Lagrangians. In \cite{KS} it is conjectured that for any graded commutative ring $k,$ quasi-equivalence classes of $A_{\infty}$-pre-categories over $k$ are in bijection with quasi-equivalence classes of $A_{\infty}$-categories over $k$ with strict (or weak) identity morphisms. In this paper we prove this conjecture for essentially small $A_{\infty}$-(pre-)categories, in the case when $k$ is a field. In particular, it follows that we can replace Fukaya $A_{\infty}$-pre-category with a quasi-equivalent actual $A_{\infty}$-category. We also present natural construction of pre-triangulated envelope in the framework of $A_{\infty}$-pre-categories. We prove its invariance under quasi-equivalences.