This is a follow-up to my question about nilpotent orbits here asked in connection with an earlier discussion of symplectic resolutions. Leaving aside the connections with algebraic geometry and representation theory, I'm curious about what has been written down explicitly for the exceptional Lie algebras $E_6, E_7, E_8$.

Start with a simple Lie algebra $\mathfrak{g}$ of rank $n$ over $\mathbb{C}$ (or the Lie algebra of a simple algebraic group $G$ over any algebraically closed field of good characteristic). The finitely many nilpotent orbits in $\mathfrak{g}$ (or unipotent classes in $G$) have been well studied, with expositions for example in the books by Carter (1985) and Collingwood-McGovern (1993). Carter provides detailed tables in 13.1 and partial order graphs in 13.4 (though the $E_7, E_8$ graphs have a few missing edges compared to the correct pictures in Spaltenstein's lecture notes). In particular, it's easy for exceptional types to compare the total number of orbits, labelled by Dynkin or Bala-Carter, with the number of *special* orbits in Lusztig's sense.

While the special orbits are defined only indirectly via several kinds of representation theory, all *Richardson orbits* are special. The converse starts to fail in type $D_4$, where there is one special but non-Richardson orbit. Richardson orbits belong to one or more of the $2^n$ conjugacy classes of parabolic subalgebras in $\mathfrak{g}$; for instance, the subregular nilpotent orbit intersects the nilradical of each minimal parabolic $\mathfrak{p}$ in its dense orbit under the corresponding adjoint subgroup $P$. In type $F_4$ there are 16 orbits, 11 special and 9 of these Richardson.

Has anyone computed the full list of Richardson orbits in types $E_6, E_7, E_8$?

As noted earlier, Hirai's paper here gives "generators and relations" for producing such a list, along with explicit information about Richardson orbits with "non-even" Dynkin labelling. But for example, what is the list of Richardson orbits coming from the 256 classes of parabolics for $E_8$? One knows that there are 70 orbits, 46 of which are special; which of the latter are Richardson?