Suppose that $X$ is a quasi-projective variety over a field $k$ and that we further know that for every coherent sheaf $\mathcal{F}$, $H^i(X,\mathcal{F})$ is finitely generated over $\Gamma(O_X)$. Is there an example where the induced morphism $X \to$ Spec $\Gamma(O_X)$ is not proper?

As the contributer a-fortiori notes in the comments to this question Is $H^i(X,F)$ finitely generated over $\Gamma(O_X)$ if $F$ is coherent?, there is no such example if all the groups $H^i(X,\mathcal{F})$ are known to be finitely generated over $k$. Not being strong in algebraic geometry, I can't off-hand tell whether his argument can be generalized.