The two-dimensional Jacobian Conjecture over $\mathbb{C}$ says the following: Let $p,q \in \mathbb{C}[x,y]$ satisfy $\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in \mathbb{C}-\{0\}$. Then $\mathbb{C}[p,q]=\mathbb{C}[x,y]$.
Let $\beta: \mathbb{C}[x,y] \to \mathbb{C}[x,y]$ be the following involution (= automorphism of order two): $x \mapsto x, y \mapsto -y$.
Then we can write: $p=s_1+k_1$ and $q=s_2+k_2$, where $s_1,s_2$ are symmetric w.r.t. $\beta$ and $k_1,k_2$ are skew-symmetric w.r.t. $\beta$.
It is well-know and was proved in several ways that $\mathbb{C}[p,q] \subseteq \mathbb{C}[x,y]$ is flat.
Is $\mathbb{C}[s_1,k_1,s_2,k_2] \subseteq \mathbb{C}[x,y]$ flat?
I am not sure if one of the known proofs for flatness of $\mathbb{C}[p,q] \subseteq \mathbb{C}[x,y]$ can be adjusted here.
Now I have also asked the above question in MO.
Any hints and comments are welcome! Thank you.