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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 $α$ and X-ray photons in the presence of excess radio background at cosmic dawn. In extreme cases, the Ly $α$ 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 $α$ 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.