Panyushev's orbit conjecture for antichains above the simple roots

Let $\Delta$ be a reduced irreducible root system with positive roots $\Delta^+$, simple roots $\Pi$, Coxeter number $h$, and reverse operator $\mathfrak X_0$ on the antichains of the subposet $\Delta^+\setminus\Pi$. For an orbit $\mathcal O$, write $\#\Gamma$ for the cardinality of an antichain $\Gamma$. Panyushev's orbit conjecture. (i) If $w_0=-1$, then $\operatorname{ord}(\mathfrak X_0)=h-1$. (ii) If $w_0\ne -1$, then $\mathfrak X_0^{h-1}$ is the involution induced by $-w_0$ and $\operatorname{ord}(\mathfrak X_0)=2h-2$. (iii) For every $\mathfrak X_0$-orbit $\mathcal O$,

$$\frac{1}{\#\mathcal O}\sum_{\Gamma\in\mathcal O}\#\Gamma=\frac{\#(\Delta^+\setminus\Pi)}{h-1}=\frac n2\cdot\frac{h-2}{h-1}.$$

The statement predicts the analogous orbit-order and average-size properties after removing the simple roots; the supplied text reports verification in several low-rank and exceptional cases but gives no general resolution.