Classification of nilpotent orbits over local fields (for type ABCD via partitions )

Question

Let $\mathfrak g$ be a simple Lie algebra over a char $0$ local field $F$ (e.g. $F=\mathbb R$ or $F=\mathbb Q_p$) with its adjoint group $G$. Let $\mathcal N \subseteq \mathfrak g$ be its nilpotent variety over $F$ with the natural adjoint $G$-action.

We know several classifications of nilpotent orbits over $F=\mathbb C$: Bala-Carter (Levis and distinguished parabolic subalgebras), Dynkin–Kostant (a subset of weighed Dynkin diagrams) via Jacobson–Morozov theorem, and partitions (only for type $A, B, C, D$) via Jordan normal forms.

Now the work of George J. McNinch on Nilpotent orbits over ground fields of good characteristic (https://arxiv.org/abs/math/0209151) shows for any local field $F$ that there are only finitely many nilpotent orbits in $\mathcal{N}(F)$ (essentially because of the finiteness of $H^1(F,-)$ of stabilizers). So a classification may be possible.

Here are my questions: let $\mathfrak g$ be of type $A, B, C, D$,

1. could we classify nilpotent orbits over $F$ by some enhanced partitions?
2. Given a complex nilpotent orbit, how many $F$-nilpotent orbits are in it?
3. Assume $F$ is $p$-adic. Could we classify $F$-nilpotent orbits that could be defined over $O_F$?

This may be regarded as a question of stable conjugacy for nilpotent elements in endoscopy. I believe this might be well-known by computing $H^1$ of stabilizers of nilpotent orbits.

Recall over $\mathbb C$

1. for type $A_n$ ($\mathfrak g=sl_{n+1}$), nilpotent orbits are in bijection with partitions of $n + 1$.
2. for type $B_n$ ($\mathfrak g=so_{2n+1}$), nilpotent orbits are in bijection with partitions of $2n + 1$ where even parts occur with even multiplicity;
3. for type $C_n$ ($\mathfrak g=sp_{2n}$), nilpotent orbits are in bijection with partitions of $2n$ where odd parts occur with even multiplicity.

Motivation: the case $F=\mathbb R$ is interesting. For $sl_2(\mathbb R)$, nilpotent orbits are given by $0$, $\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ and $\begin{pmatrix} 0 & -1 \\ 0 & 0 \end{pmatrix}$. There is a real Bala-Carter classification by A. G. No ̈el, Nilpotent orbits and theta-stable parabolic Subalgebras, AMS Journal of representation theory 2, (1998), 1-32. The proof is reduced to a graded case over $\mathbb C$ for symmetric spaces via Kostant-Sekiguchi correspondence, and there exists a partition type description (Section 4 of loc. cit.). I do not know any $p$-adic Bala-Carter theory.