The following is a known result in algebraic geometry:

Let $k$ be an algebraically closed field of characteristic zero (for example, $k=\mathbb{C}$). Let $L$ be a field such that $k \subset L \subset k(x,y)$ and $L$ is of transcendence degree two over $k$. Then there exist $h_1,h_2 \in k(x,y)$ such that $L=k(h_1,h_2)$.

Is it possible to find $g_1,g_2 \in k[x,y]$ such that $L=k(g_1,g_2)$?

The motivation is the following result: If $k \subset L \subset k(x,y)$ is of transcendence degree one over $k$, then $L=k(h)$, where $h \in k[x,y]$; see this answer. Perhaps the arguments in that answer are also applicable here?

I have asked the above question in MSE, with no comments. See also this question.

Thank you very much!