Let $A,B$ be $k$-algebras with $k$ a commutative ring. Suppose that the free product $A \cdot_k B$ is a free associative $k$-algebra of rank $n$. Does it imply that $A,B$ are both free associative $k$-algebras?
Working this sort of "factorization" problems in several different settings, I realized that there is no mention in the literature (which I have access) in this case and I am not able to prove it.
EDIT: The question is false if is considered the usual tensor product.