Hot thermal universe endowed with massive dark vector fields and the Hubble tension

The value of Hubble constant inferred from Planck measurements of anisotropies in the cosmic microwave background is at $4.4\sigma$ tension with direct astronomical measurements at low redshifts. Very recently, it has been conjectured that this discrepancy may be reconciled if a small fraction of the dark matter is described by three mutually orthogonal vector fields of the same mass. We study the thermal description of this model and use the observationally-inferred primordial fractions of baryonic mass in $^4{\rm He}$ to constrain its phase space. We show that while the sterile vector fields may help to alleviate a little bit the existing tension in the measurements of the Hubble parameter, they cannot eliminate the discrepancy between low- and high-redshift observations.