vector bundles on the affine line over a PID

Question

Let $R$ be a PID. Is every finitely generated projective $R[T]$-module free? In other words, is every vector bundle on $\mathbb{A}^1_R$ trivial?

For $R=k[X]$ this is true by the Theorem of Quillen-Suslin. If it fails in general, what happens for $k[X,X^{-1}]$?

Answer 1

Quote from Lam's Serre's problem on projective modules:

In 1958, Seshadri showed that Serre's conjecture is true for two variables (i.e. for $A = k[x_1,x_2]$). In fact, Seshadri proved that f.g. projectives over R[t] are free if $R$ is any commutative PID.

Reference: Seshadri, C.S., Triviality of vector bundles over the affine space $K^2$, Proc. Nat. Acad. Sci. USA 44, 456-458.