Multiplicity of Galactic Cepheids from long-baseline interferometry~III. Sub-percent limits on the relative brightness of a close companion of $delta$~Cephei

We report new CHARA/MIRC interferometric observations of the Cepheid archetype $delta$ Cep, which aimed at detecting the newly discovered spectroscopic companion. We reached a maximum dynamic range $Delta H $ = 6.4, 5.8, and 5.2 mag, respectively within the relative distance to the Cepheid $r < 25$ mas, $25 < r < 50$ mas and $50 < r < 100$ mas. Our observations did not show strong evidence of a companion. We have a marginal detection at $3sigma$ with a flux ratio of 0.21%, but nothing convincing as we found other possible probable locations. We ruled out the presence of companion with a spectral type earlier than F0V, A1V and B9V, respectively for the previously cited ranges $r$. From our estimated sensitivity limits and the Cepheid light curve, we derived lower-limit magnitudes in the $H$ band for this possible companion to be $H_mathrm{comp} > 9.15, 8.31$ and 7.77 mag, respectively for $r < 25$ mas, $25 < r < 50$ mas and $50 < r < 100$ mas. We also found that to be consistent with the predicted orbital period, the companion has to be located at a projected separation $< 24$ mas with a spectral type later than a F0V star.