  The tight radial acceleration relation (RAR) obeyed by rotationally supported
disk galaxies is one of the most successful a priori predictions of the
modified Newtonian dynamics (MOND) paradigm on galaxy scales. Another important
consequence of MOND as a classical modification of gravity is that the strong
equivalence principle (SEP) -- which requires the dynamics of a small,
free-falling, self-gravitating system not to depend on the external
gravitational field in which it is embedded -- should be broken. Multiple
tentative detections of this so-called external field effect (EFE) of MOND have
been made in the past, but the systems that should be most sensitive to it are
galaxies with low internal gravitational accelerations residing in galaxy
clusters within a strong external field. Here, we show that ultra-diffuse
galaxies (UDGs) in the Coma cluster do lie on the RAR, and that their velocity
dispersion profiles are in full agreement with isolated MOND predictions,
especially when including some degree of radial anisotropy. However, including
a breaking of the SEP via the EFE seriously deteriorates this agreement. We
discuss various possibilities to explain this within the context of MOND,
including a combination of tidal heating and higher baryonic masses. We also
speculate that our results could mean that the EFE is screened in cluster UDGs.
The fact that this would happen precisely within galaxy clusters, where
classical MOND fails, could be especially relevant to the nature of the
residual MOND missing mass in clusters of galaxies.
