In his 2004 paper "ad-nilpotent ideals of a Borel subalgebra: generators and duality" (https://www.sciencedirect.com/science/article/pii/S0021869303006380), Panyushev conjectured (Conjecture 6.1) the existence of a certain natural involution on the set of antichains of the root poset $\Phi^+$ of an irreducible crystallographic root system $\Phi$. The main property desired is that an antichain of cardinality $k$ is sent to an antichain of cardinality $n-k$, where $n$ is the number of simple roots of $\Phi$. In that paper he constructs the desired involution for Type A, and Type B/C.

Question: Have there been any updates/progress on this conjecture?


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