Strict-unit replacement conjecture for $A_{\infty}$-pre-categories

Let an $A_{\infty}$-pre-category be understood with the equivalence relation defined in the source, and let an $A_{\infty}$-category with strict identity morphisms be a non-unital $A_{\infty}$-category with strict units. Strict-unit replacement conjecture. The equivalence classes of $A_{\infty}$-pre-categories are in one-to-one correspondence with the equivalence classes of $A_{\infty}$-categories with strict identity morphisms. This claims that the pre-category formalism and the strict-unit formalism classify the same objects up to equivalence; the source gives no resolution.