Conjecture on the leading weight of functions on a finite nilpotent-orbit cover

Let ${^{\scriptscriptstyle L\!}}{\mathcal O}$ be a nilpotent orbit in the Lie algebra of the Langlands dual group, let $\mathcal O=d({^{\scriptscriptstyle L\!}}{\mathcal O})$ be the dual nilpotent orbit, and let $\widetilde{\mathcal O}=G/K$ be the associated finite cover. Write the $G$-module of global functions as

$$\mathbb C[\widetilde{\mathcal O}]=\sum_{\lambda\in\Lambda_+}m_\lambda\operatorname{Ind}_T^G\lambda,$$

and let $\mu$ be a weight of largest length with $m_\mu\ne0$. Leading-weight conjecture. The weight $\mu$ is unique and is the Dynkin weight of ${^{\scriptscriptstyle L\!}}{\mathcal O}$. This is a more concrete formulation for the associated finite cover: it predicts that the highest-length induced representation occurring in its coordinate ring is uniquely determined by the Dynkin weight of the Langlands-dual orbit. The source does not give a resolution, so the claim remains open.