A result of E. Formanek, in its two-dimensional version, says:

Let $k$ be a field of characteristic zero and let $R=k[x,y]$ be the polynomial ring in two variables. Let $p,q \in R$ have invertible Jacobian in $R$ (namely, the Jacobian is a non-zero scalar). If there exists $w \in R$ such that $k[p,q][w]=R$, then $k[p,q]=R$.

Now let $I$ be the ideal of $R$ generated by $p,q$: $I=\langle p,q \rangle= Rp+Rq$.

Is it possible to prove the following claim by adjusting Formanek's proof:

If there exists $w \in R$ such that $k[p,q][w]+I=R$, then $k[p,q]=R$.

Of course, if $I \subseteq k[p,q][w]$, then this is just Formanek's result.

I have asked the above question in MSE. This is a relevant question.

Thank you very much!