Cancellation conjecture for free associative algebras

Let $R$ be an algebra over a field $K$, let $z$ be a free generator, and let $x_1,\dots,x_n$ be noncommuting free generators. Write $K\langle x_1,\dots,x_n\rangle$ for the free associative $K$-algebra on these generators, and $R\ast K[z]$ for the free product. Cancellation conjecture for free associative algebras. If

$$R\ast K[z]\cong_K K\langle x_1,\dots,x_n\rangle,$$

then

$$R\cong_K K\langle x_1,\dots,x_{n-1}\rangle.$$

The source proves this conjecture for free associative algebras of rank two; the supplied text does not state its status in other ranks.