A question motivated by If a polynomial ring is a free module over some subring, is that subring itself a polynomial ring? and If a polynomial ring is finite free over a subring, is the subring polynomial?.

Let $k$ be an algebraically closed field of characteristic zero (for simplicity), and let $A$ be a $k$-subalgebra of a polynomial ring $B := k[x_1, \dotsc, x_n]$ such that $B$ is a flat $A$-module of finite type.

Then, must $A$ also be a polynomial ring? It's true if $n=2$ (see Regular subrings of a polynomial ring by Masayoshi Miyanishi for a proof).

If not, could we classify all such $A$ ?