  The wavefront set is a fundamental invariant of an admissible representation
arising from the Harish-Chandra-Howe local character expansion. In this paper,
we give a precise formula for the wavefront set of an irreducible
representation of real infinitesimal character in Lusztig's category of
unipotent representations in terms of the Deligne-Langlands-Lusztig
correspondence. Our formula generalizes the main result of arXiv:2112.14354,
where this formula was obtained in the Iwahori-spherical case. We deduce that
for any irreducible unipotent representation with real infinitesimal character,
the algebraic wavefront set is a singleton, verifying a conjecture of M\oeglin
and Waldspurger. In the process, we establish new properties of the generalized
Springer correspondence in relation to Lusztig's families of unipotent
representations of finite reductive groups.
