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.

In another article of AHR, 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, 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!

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 Conjectures on tilting modules and antispherical $p$-cells of Achar–Hardesty–Riche.

In another article On the Humphreys conjecture on support varieties of tilting modules of AHR, 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 Integral exotic sheaves and the modular Lusztig-Vogan bijection, AHR show that the 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?