Bass–Quillen conjecture on projective modules over polynomial rings

Let $A$ be a regular ring of finite Krull dimension, and let $n\ge 1$. A finitely generated projective module over $A[x_1,\ldots,x_n]$ is extended from $A$ if it is obtained from a finitely generated projective $A$-module by extension of scalars along $A\to A[x_1,\ldots,x_n]$. Bass–Quillen conjecture. Every finitely generated projective module over $A[x_1,\ldots,x_n]$ is extended from $A$ for any $n\ge 1$. This is the classical Bass–Quillen conjecture on homotopy invariance for finitely generated projective modules; the dimension-$2$ case is known in the setting described in the paper, while the conjecture in full generality remains open.