Quasi-admissible, raisable nilpotent orbits and covering Barbasch-Vogan duality

For simply-connected Lie groups of type E over $p$-adic local field $F$, we determine the degree of the cover required for a given $F$-split nilpotent orbit to be quasi-admissible or raisable, respectively. Combining this result with the previously computed data for other types by Gao-Liu-Tsai, we prove that all $F$-split nilpotent orbits whose geometry type contained in the image of the covering Barbasch-Vogan duality map $d_{\mathrm{BV},G}^{(n)}$ of almost-simple Lie groups $G$ in each Cartan type are always$\overline{G}^{(n)}$-quasi-admissible.