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In the paper Affine varieties dominated by $\mathbf{C}^2$ 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.