Using spectroscopic redshifts of high-redshift z>1.5 cluster galaxies I computed their location in the projected velocity-versus-position phase-space to separate the cluster sample into a virialized region of objects accreted longer ago and a region of infalling galaxies. I used near-infrared spectroscopy of Halpha, [NII], [OIII] and Hbeta of galaxies distributed over a region that corresponds to about one virial radius, and found the computed gas metallicities O/H of these high-redshift cluster galaxies inside half R200 to be enhanced compared to infalling galaxies and field galaxies at similar redshifts and of similar masses. This indicates that the density of the intracluster medium in this massive cluster environment becomes high enough toward the cluster center such that the ram pressure exceeds the restoring pressure of the hot gas reservoir of cluster galaxies. This can remove the gas reservoir stopping the inflow of diluting gas thereby enhancing gas metallicities and initiating the quenching of star formation; although the galaxies continue to form stars, albeit at slightly lower rates, using the available cold gas in the disk which is not stripped. Active galactic nuclei (AGN) identified from an emission line diagnostic diagram were used to test a scenario in which AGN feedback can accelerate the quenching of star formation by acting on the (cold) gas deepest in the galaxy's potential well, which is the most difficult gas for ram pressure stripping to remove.