In studies of accreting black holes in binary systems, empirical relations have been proposed to quantify the coupling between accretion processes and ejection mechanisms. These processes are probed, respectively, by means of X-ray and radio/optical-infrared observations. The relations predict, given certain accretion conditions, the expected energy output in the form of a jet. We investigated this coupling by studying the black hole candidate Swift J1753.5-0127, via multiwavelength-coordinated observations over a period of ~4 yr. We present the results of our campaign showing that, all along the outburst, the source features a jet that is fainter than expected from the empirical correlation between the radio and the X-ray luminosities in a hard spectral state. Because the jet is so weak in this system the near-infrared emission is, unusually for this state and luminosity, dominated by thermal emission from the accretion disc. We briefly discuss the importance and the implications of a precise determination of both the slope and the normalization of the correlations, listing some possible parameters that broad-band jet models should take into account to explain the population of sources characterized by a dim jet. We also investigate whether our data can give any hint on the nature of the compact object in the system, since its mass has not been dynamically measured.