Distinct central characters for cuspidal pairs on a nilpotent orbit

Let $G$ be any connected reductive group and suppose that $\boldsymbol{k}$ is big enough for $G$. For a fixed nilpotent orbit $\mathscr{O}$, consider the cuspidal pairs $(\mathscr{O},\mathcal{E})\in\mathfrak{N}_{G,\boldsymbol{k}}^{\mathrm{cusp}}$ involving that orbit. Distinct-central-character conjecture. These cuspidal pairs have distinct central characters; equivalently, each nilpotent orbit supports at most one cuspidal pair with a given central character. This is presented as a weaker form of the preceding theorem and is known in types $A$--$D$, $G_2$, and $E_6$, while the general case is left open.