In "Unipotent support for irreducible representations", Lusztig proves the existence of a unique maximal unipotent class $\mathscr{O}_\pi$, called unipotent support, where the character of a representation $\pi$ is not trivial.

I would like first to ask if the relationship between the Lusztig correspondence $\mathfrak{L}$ and the unipotent support is known. More precisely, take two representations $\rho$ and $\sigma$ in the same Lusztig series such that $\mathscr{O}_\rho\subset\overline{\mathscr{O}_\sigma}$ (this defines the so-called "closure order"). Can something be said about the unipotent support of $\mathfrak{L}(\rho)$ and $\mathfrak{L}(\sigma)$? For instance, would one be contained in the closure of the other?

Also, it is known that unipotent representations of unitary groups are parametrized by partitions of the order of the group. Is there a combinatorial way to determine the unipotent support of such a representation in terms of the corresponding partition? I know that for symplectic groups there is indeed a combinatorial way.

Lastly, is there a package in Magma or GAP that calculates the unipotent support for (small) finite groups of Lie type?