Directed $A_{\infty}$-category correspondence conjecture

Let a directed $A_{\infty}$-pre-category be an $A_{\infty}$-pre-category whose objects are indexed by ${{\bf Z}}$ with transversal sequences ordered by the indices, and consider $A_{\infty}$-categories with strict identity morphisms and countable class of objects. Directed $A_{\infty}$-category correspondence conjecture. Equivalence classes of directed $A_{\infty}$-pre-categories are in one-to-one correspondence with equivalence classes of $A_{\infty}$-categories with strict identity morphisms and countable class of objects. This is the directed version of the preceding strict-unit replacement claim; the source gives no resolution.