Conjecture on the maximal Whittaker orbit of a genuine representation of a covering general linear group

Consider the orbit

$$\mathcal{O}^{r,n}:=(n_\alpha^a b)$$

of ${\rm GL}_r$, where $r=a\cdot n_\alpha+b$ and $0\leqslant b<n_\alpha$. Let $\overline{{\rm GL}}_r^{(n)}$ be the $n$-fold covering group and let $\pi\in {\rm Irr}_{\rm gen}(\overline{{\rm GL}}_r^{(n)})$. Write $\mathcal{N}_{\rm Wh}^{\rm \max}(\pi)$ for the set of maximal nilpotent orbits occurring in the Whittaker support of $\pi$, ordered by dominance.

Maximal Whittaker-orbit conjecture. One has $\mathcal{O}^{r,n}\leqslant\mathcal{O}$ for every $\mathcal{O}\in\mathcal{N}_{\rm Wh}^{\rm \max}(\pi)$.

This predicts a uniform lower bound for the maximal Whittaker orbits of genuine representations of covering general linear groups. The supplied text presents it as a belief, and gives no evidence of a resolution.