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A theorem of Mœglin-Waldspurger for covering groups

Let $E$ be a non-Archimedian local field of characteristic zero and residue characteristic $p$. Let ${\bf G}$ be a connected reductive group defined over $E$ and $π$ an irreducible admissible representation of $G={\bf G}(E)$. A result of C. Mœglin and J.-L. Waldspurger (for $p \neq 2$) and S. Varma (for $p=2$) states that the leading coefficient in the character expansion of $π$ at the identity element of ${\bf G}(E)$ gives the dimension of a certain space of degenerate Whittaker forms. In this paper we generalize this result of Mœglin-Waldspurger to the setting of covering groups $\tilde{G}$ of $G$.