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We test if Refracted Gravity (RG) can describe the dynamics of disk galaxies without resorting to dark matter. RG is a classical theory of gravity where the standard Poisson equation is modified by the gravitational permittivity, $ε$, a universal monotonic function of the local mass density. We use the rotation curves and the vertical velocity dispersions of 30 galaxies in the DiskMass Survey (DMS) to determine $ε$. RG describes the kinematic profiles with mass-to-light ratios consistent with SPS models, and disk thicknesses in agreement with observations, once observational biases are considered. Our results rely on setting the three free parameters of $ε$ for each galaxy. However, we show that the differences of these parameters from galaxy to galaxy could be ascribed to statistical fluctuations. We adopt an approximate method to find a single set of parameters that may properly describe the kinematics of the entire sample and suggest the universality of $ε$. Finally, we show that the RG models of the individual rotation curves can only partly describe the radial acceleration relation (RAR). Evidently, the RG models underestimate the observed accelerations of 0.1-0.3 dex at low Newtonian accelerations. Another problem is the strong correlation, at much more than 5$σ$, between the residuals of the RAR models and three radially-dependent properties of galaxies, whereas the DMS data show a considerably less significant correlation, at more than 4$σ$, for only two of them. These correlations might originate the non-null intrinsic scatter of the RG models, at odds with the observed intrinsic scatter of galaxy samples different from DMS, which is consistent with 0. Further studies are required to assess if these discrepancies in the RAR originate from the DMS sample, which might not be ideal for deriving the RAR, or if they are genuine failures of RG.