  Apart from its anomalously large depth, the cosmological 21-cm absorption
signal measured by the EDGES collaboration also has a shape that is distinctly
different from theoretical predictions. Models with non-traditional components
such as super-adiabatic baryonic cooling or an excess radio background explain
the depth of the observed profile, but still conspicuously fail to explain its
shape. In this paper, we quantify the requirements imposed by the EDGES
measurement on sources of Ly $\alpha$ and X-ray photons in the presence of
excess radio background at cosmic dawn. In extreme cases, the Ly $\alpha$ and
X-ray emissivities require to be enhanced by up to an order of magnitude
relative to traditional models. Furthermore, this enhancement needs to be
active only for a short duration. We find that under conventional assumptions
for the cosmic star formation rate density, standard stellar populations are
incapable of meeting these conditions. Only highly unusual models of massive
metal-free stars seem to provide a possible mechanism. Conversely, if the
sources of Ly $\alpha$ and X-ray photons are compelled to have standard
properties, the EDGES measurement puts strong demands on the cosmic star
formation rate density. This provides interesting falsifiable predictions for
high-redshift galaxy surveys enabled by \textit{James Webb Space Telescope}
(\textit{JWST}). We derive predictions for galaxy UV luminosity functions and
number densities, and show that a deep \textit{JWST} survey with a limiting UV
magnitude of $m_\mathrm{UV,lim}=32$ would potentially be able to rule out the
predictions enforced by the EDGES measurement.
