  We study a conjectural formula for the maximal elements in the wavefront set
associated with a theta representation of a covering group over $p$-adic
fields. In particular, it is shown that the formula agrees with the existing
work in the literature for various families of groups. We also recapitulate the
results of an analogous formula in the archimedean case, which motivated the
conjectural formula in the $p$-adic setting.
