A result of [E. Formanek][1], 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 $p,q$:
$I=\langle p,q \rangle= Rp+Rq$.

**Question**. 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][2].

[This][3] is a relevant question.

Thank you very much!


  [1]: https://www.researchgate.net/publication/227101632_Two_notes_on_the_Jacobian_Conjecture
  [2]: https://math.stackexchange.com/questions/4060690/concerning-a-result-of-formanek-about-kf-1-ldots-f-nf-n1-kx-1-ldots-x
  [3]: https://mathoverflow.net/questions/366216/question-about-jacobian-subalgebra