In the paper [*Affine varieties dominated by $\mathbf{C}^2$*][1] Gurjar settles the case $n=2$. More precisely, he proves that if $\mathrm{X}$ is an affine normal variety such that there exists a proper surjective map $\mathbf{A}^2\rightarrow\mathrm{X}$, then every vector bundle on $\mathrm{X}$ is trivial.


  [1]: https://eudml.org/doc/139833