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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?

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    $\begingroup$ Did you see arxiv.org/abs/2106.04268? They claim it is done for $p \ge h$, which looks like the full conjecture to me. (One would have to ask the authors if they expect something interesting for small p.) $\endgroup$ Commented Jul 31, 2023 at 7:07

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The paper Silting complexes of coherent sheaves and the Humphreys conjecture by Achar and Hardesty proves this conjecture in full generality (for $p \ge h$).

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  • $\begingroup$ Thank you very much for the answer. I will check the article and ask the authors about the idea on further conjecture for small $p$. $\endgroup$
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    Commented Aug 2, 2023 at 18:23

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