Integrality of central characters for FC sets

Let $\mathfrak{g}$ be an FC set, let $\mathsf{C}in\br{\mathfrak{g}}$ be a class, and let $alphainmathfrak{g}$. Its central character is denoted by $cech{\mathsf{C}}{\alpha}$.

Integrality conjecture. $cech{\mathsf{C}}{\alpha}$ is always an algebraic integer.

For local FC sets this follows from the corresponding result in finite-group character theory; the claim proposes that the same integrality holds for generic FC sets.