Zariski's cancellation conjecture for polynomial algebras

Let $R$ be an algebra over a field $K$, let $z$ be an indeterminate, and let $x_1,\dots,x_n$ be commuting indeterminates. Zariski's cancellation conjecture. If

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

then

$$R\cong K[x_1,\dots,x_{n-1}].$$

The conjecture is proved for $n=2$ and $n=3$ under the results described in the source, including arbitrary fields when $n=3$; for $n\geq 4$, it remains open to the authors' knowledge.