Extra-special orbit and Lagrangian-ideal conjecture

In the extra-special case, let $\mathfrak X_{\Delta(1)}$ be the reverse operator on the associated weight poset, let $\Pi_l$ be the set of long simple roots, and let $h$ be the Coxeter number.

Extra-special orbit conjecture. The number of $\mathfrak X_{\Delta(1)}$-orbits is $\#\Pi_l$, every orbit has size $h-1$, and, if $h$ is even, every orbit contains a unique Lagrangian upper ideal.

The orbit count and orbit size are suggested by the proved enumeration in the extra-special case and explicit calculations; the statement is verified in several types but remains open in general.