Orlik's orbit-average conjecture for reverse operators on 1-standard gradings

Let $\mathfrak g=\bigoplus_{i\in\mathbb Z}\mathfrak g(i)$ be a $1$-standard $\mathbb Z$-grading, so that $\mathfrak g(1)$ is a simple $\mathfrak g(0)$-module and the semisimple part of $\mathfrak g(0)$ is $\widetilde{\mathfrak g(0)}$. Let $\Delta(1)$ be the associated weight poset, let $\mathfrak X_{\Delta(1)}$ be its reverse operator, and set

$$d_1=\max\{{\mathsf{ht}}(\gamma)\mid\gamma\in\Delta(1)\}.$$

Orlik's orbit-average conjecture. The following assertions hold: (i) ${\mathsf{ord}}(\mathfrak X_{\Delta(1)})=d_1+1$; (ii) the average size of the antichains in every $\mathfrak X_{\Delta(1)}$-orbit is

$$\frac{\dim\mathfrak g(1)}{d_1+1}=\frac{\#\Delta(1)}{{\mathsf{ord}}(\mathfrak X_{\Delta(1)})};$$

and (iii) the average size of the upper ideals in every $\mathfrak X_{\Delta(1)}$-orbit is $\dim\mathfrak g(1)/2$.

These properties are supported by computations and are established in several special classes, but the general 1-standard case remains open.