The self-binding energy and stability of a galaxy in MOND-based gravity are curiously decreasing functions of its center-of-mass acceleration (of the order of 10-12 to 10-10 m s-2) toward neighboring mass concentrations. A tentative indication of this breaking of the strong equivalence principle in field galaxies is the RAVE-observed escape speed in the Milky Way. Another consequence is that satellites of field galaxies will move on nearly Keplerian orbits at large radii (100-500 kpc), with a declining speed below the asymptotically constant naive MOND prediction. But the consequences of an environment-sensitive gravity are even more severe in clusters, where member galaxies accelerate fast; no dark halo-like potential is present to support galaxies, meaning that extended axisymmetric disks of gas and stars are likely unstable. These predicted reappearances of asymptotic Keplerian velocity curves and disappearances of "stereotypic galaxies" in clusters are falsifiable with targeted surveys.