Panyushev's conjectured duality for root poset antichains

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?