de Graaf's characteristic-polynomial factorization conjecture for simple Lie algebras

Let $\mathfrak{g}$ be a complex simple Lie algebra, let $V$ be an irreducible $\mathfrak{g}$-module with highest weight $\lambda$, and let $\Pi(\lambda)$ denote the set of weights of $V$. Let $\mu_1,\dots,\mu_s$ be the distinct dominant weights occurring in $V$, let $\mathscr{W}$ be the Weyl group, and let $k_i$ be the size of the $\mathscr{W}$-orbit of $\mu_i$. Write $m_\lambda(\mu_i)$ for the multiplicity of $\mu_i$ in $V$. de Graaf's conjecture. The characteristic polynomial satisfies

$$f_{\phi}=\prod_{\mu_i\in\Pi(\lambda)}(f_{\mu_i})^{m_\lambda(\mu_i)},$$

where $f_{\mu_i}$ is an irreducible polynomial of degree $k_i=\lvert O_{\mu_i}\rvert$ obtained from the orbit $O_{\mu_i}$ of $\mu_i$. This conjecture proposes a factorization of generalized characteristic polynomials according to the dominant weights and Weyl-group orbits of an irreducible module; the supplied text gives no resolution status.