Integrality conjecture for Fourier transforms of nilpotent coadjoint orbits

Let $G$ be a reductive Lie group, let $\mathfrak{g}$ be its Lie algebra, and let $\mathcal{O}\subset \mathfrak{g}^*$ be a nilpotent coadjoint orbit. Let $H\subset G$ be a Cartan subgroup, let $\mathfrak{h}'=\operatorname{Lie}(H)'$, and let $C\subset \mathfrak{h}'$ be a connected component of the regular set. Let $\Delta$ be the roots of $\mathfrak{g}_{\mathbb{C}}$ with respect to $\mathfrak{h}_{\mathbb{C}}$, choose a set of positive roots $\Delta^+$, and let $\pi$ be the product of the roots in $\Delta^+$. The author's integrality conjecture.

$$\pi\widehat{\mathcal{O}}|_C\in \mathbb{Z}[\Delta].$$

The conjecture asks whether the Fourier transforms of nilpotent coadjoint orbits satisfy an integrality condition analogous to the integer linear combinations appearing in the paper's formula. A related result states that leading terms of irreducible character expansions are nonnegative linear combinations of Fourier transforms of nilpotent coadjoint orbits; the conjecture asserts integrality after multiplication by the product of positive roots.