Humphreys conjecture describes the support variety of tilting modules using the correspondence between two-sided cells and nilpotent orbits. But the support variety can also be given by Lusztig-Vogan bijection, as in [the work of Achar-Hardesty-Riche][1].

In [another article of AHR][2], the authors have proved that Humphreys conjecture is true when the characteristic of the base field of the algebraic group is sufficiently large. Then in [this article][3], AHR shows that Lusztig-Vogan bijection is independent of the characteristic of the base field, under certain assumptions.

It seems that the proof of Humphreys conjecture is almost done after the contributions of Achar-Hardesty-Riche. So what is left to do to complete the proof?

Thank you very much!

  [1]: https://arxiv.org/pdf/1812.09960.pdf
  [2]: https://arxiv.org/pdf/1707.07740.pdf
  [3]: https://arxiv.org/pdf/1810.08897.pdf