Equivalence of unitality and homotopy unitality for $A_\infty$-morphisms

Let ${\mathsf f}:A_1,\dots,A_n\to B$ be an $A_\infty$-morphism between unital $A_\infty$-algebras. It is unital when, for every $1\leq k\leq n$, the cycles ${\mathsf i}^{A_k}{\mathsf f}_{e_k}$ and ${\mathsf i}^B$ differ by a boundary. A homotopy unital structure on ${\mathsf f}$ is a choice of a strictly unital extension ${\mathsf f}^+:A_1^+,\dots,A_n^+\to B^+$ satisfying the stated compatibility conditions.

Unitality–homotopy unitality equivalence. An $A_\infty$-morphism is unital if and only if it admits a homotopy unital structure; equivalently, any unital $A_\infty$-morphism admits a homotopy unital structure.

This identifies the property of unitality with the existence of the additional homotopy unital structure, showing that the two notions of unitality agree for $A_\infty$-morphisms. The supplied text gives no evidence that the assertion has been proved or disproved.