Fourier-analytic cuspidal restriction conjecture for Lie algebra orbit closures

Let $F$ be a local or finite field, let $\underline G$ be a split semisimple group with Lie algebra $\mathfrak g$, and let $\Omega\subset\mathfrak g$ be a conjugacy class with closure $\bar\Omega$. For $(x,N)\in Y^\Omega$, where $Y^\Omega$ consists of pairs satisfying $x+N\subset\bar\Omega$, let $f_y$ be the function on $N$ given by $f_y(n)=f(x+n)$ and let $\hat f_y$ be its Fourier transform on $N^\vee$. Define $\mathcal S(\bar\Omega)_{cusp}$ as the restrictions to $\bar\Omega$ of functions $f\in\mathcal S(\mathfrak g)$ whose Fourier transforms are supported on the elliptic set $\mathfrak g^\vee_{ell}$, and define $\mathcal S_w(\bar\Omega)$ by requiring

$$\hat f_y\in\mathcal S(N_y^\vee\setminus\{0\})$$

for every $y=(x,N)\in Y^\Omega$. Fourier-analytic cuspidal restriction conjecture.

$$\mathcal S(\bar\Omega)_{cusp}=\mathcal S_w(\bar\Omega).$$

This is formulated as a Lie algebra analogue of the preceding conjecture and also makes sense for archimedean fields. For finite and non-archimedean fields, the defining Fourier-support condition is equivalent to vanishing of the integral over $N_y$, while the general assertion remains open in the source.