The GLEAMing of the first supermassive black holes: III. Radio sources with ultra-faint host galaxies

We present deep near-infrared Kₛ-band imaging for 35 of the 53 sources from the high-redshift ($z > 2$) radio galaxy candidate sample defined in Broderick et al. (2022). These images were obtained using the High-Acuity Widefield K-band Imager (HAWK-I) on the Very Large Telescope. Host galaxies are detected for 27 of the sources, with Kₛ ≈ 21.6-$$23.0$ mag (2$''$ diameter apertures; AB). The remaining eight targets are not detected to a median 3σ depth of Kₛ ≈ 23.3 mag ($2''$ diameter apertures). We examine the radio and near-infrared flux densities of the 35 sources, comparing them to the known $z > 3$ powerful radio galaxies with 500-MHz radio luminosities L_500\, MHz > 10²⁷ W Hz⁻¹. By plotting 150-MHz flux density versus Kₛ-band flux density, we find that, similar to the sources from the literature, these new targets have large radio to near-infrared flux density ratios, but extending the distribution to fainter flux densities. Five of the eight HAWK-I deep non-detections have a median 3σ lower limit of Kₛ 23.8 mag ($1.5''$ diameter apertures); these five targets, along with a further source from Broderick et al. (2022) with a deep non-detection (Kₛ 23.7 mag; 3σ; $2''$ diameter aperture) in the Southern H-ATLAS Regions Kₛ-band Survey, are considered candidates to be ultra-high-redshift ($z > 5$) radio galaxies. The extreme radio to near-infrared flux density ratios (>10⁵) for these six sources are comparable to TN J0924$-$2201, GLEAM J0856$+$0223 and TGSS J1530$+$1049, the three known powerful radio galaxies at $z > 5$. For a selection of galaxy templates with different stellar masses, we show that z 4.2 is a plausible scenario for our ultra-high-redshift candidates if the stellar mass M* 1010.5\, M_. [abridged]