There is a useful special case of the notion of $A_{\infty}$-pre-category (independently a similar notion was suggested in [Se]).

The extension property is equivalent to the following one: for any object $X_{i}$ there are exist objects $X_{j},j<i$ and $X_{m},m>i$ which are quasi-isomorphic to $X$. Then one can formulate the following version of the previous conjecture.

Having an $A_{\infty}$-category ${\cal C}$ with strict identity morphisms, and countable class of objects, one can construct an infinite sequence of objects $(X_{i})_{i\in{{\bf Z}}}$ such that each objects appears infinitely many times for positive and negative $i$. Then a directed $A_{\infty}$-pre-category ${\cal C}^{\prime}$ is defined by setting $Hom_{{\cal C}^{\prime}}(X_{i},X_{j})=Hom_{{\cal C}}(X_{i},X_{j})$ for $i<j$. All other $Hom^{\prime}s$ are not defined.