The centraliser invariant-field conjecture for commuting involutions

Let ${\mathfrak g}={\mathfrak g}_0\oplus{\mathfrak g}_1$ be the decomposition associated with a commuting involution, let $\mathcal N$ be the nilpotent cone in ${\mathfrak g}$, and let $e\in{\mathfrak g}_0\cap\mathcal N$. Write ${\mathfrak g}_0^e$ and ${\mathfrak g}_1^e$ for the centralisers of $e$ in the respective summands. Let $G_0^e$ be the connected subgroup of $G_0$ with Lie algebra ${\mathfrak g}_0^e$, acting on $({\mathfrak g}_1^e)^*$, and let $\Bbbk(({\mathfrak g}_1^e)^*)^{G_0^e}$ denote the field of invariant rational functions.

Centraliser invariant-field conjecture. For any $e\in{\mathfrak g}_0\cap\mathcal N$,

$$\operatorname{trdeg}\Bbbk(({\mathfrak g}_1^e)^*)^{G_0^e}\leq\operatorname{rk}{\mathfrak g}.$$

This property would yield a case-free proof of the strange inequality for commuting involutions by combining Rosenlicht's theorem with the trivial action of the one-dimensional unipotent group generated by a nonzero nilpotent centraliser element. The source presents it as conjectural and gives no resolution.