Throughout, let $k$ be an algebraically closed field. For two $k$-algebras $A,B$ let us write $A \cong_k B$ to mean that $A,B$ are isomorphic as $k$-algebras.

It is known that if $A$ is an integral domain which is finitely generated $k$-algebra and such that $A[T] \cong_k k[X_1,...,X_n] $ , then $A \cong_k k[X_1,...,X_{n-1}]$ as $k$-algebras, when $2\le n \le 3$.

A proof for $n=2$ case can be fond in On the Uniqueness of the Coefficient Ring in a Polynomial Ring

and a proof of $n=3$ case can be found in An algebraic proof of a cancellation theorem for surfaces.

My questions are :

Are similar cancellation theorems known for Laurent polynomial or power series rings ? i.e. let $A$ be an integral domain which is a finitely generated $k$-algebra , such that $A[T,T^{-1}] \cong_k k[X_i, X_i^{-1} : 1 \le i \le n]$ (resp. $A[[T]] \cong_k k[[X_1,...,X_n]]$ ) ; then when can we say that $A\cong_k k[X_i, X_i^{-1} : 1 \le i \le n-1]$ (resp. $A \cong_k k[[X_1,...,X_{n-1}]] $) ?

Any reference towards these questions will also be very appreciated.