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](https://arxiv.org/abs/1812.09960) of Achar–Hardesty–Riche.


In another article [On the Humphreys conjecture on support varieties of tilting modules](https://arxiv.org/abs/1707.07740) 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](https://arxiv.org/abs/1810.08897), 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?



What is the remaining difficulty in the proof of the Humphreys conjecture (on the support variety of tilting modules)?
The paper [Silting complexes of coherent sheaves and the Humphreys conjecture](https://arxiv.org/abs/2106.04268) by Achar and Hardesty proves this conjecture in full generality (for $p \ge h$).


