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We explore models of massive ($\gt 10^{10}~{\rm M}_{\odot}$) satellitequenching in massive clusters at $z\gtrsim1$ using an MCMC framework, focusingon two primary parameters: $R_{\rm quench}$ (the host-centric radius at whichquenching begins) and $\tau_{\rm quench}$ (the timescale upon which a satellitequenches after crossing $R_{\rm quench}$). Our MCMC analysis shows two localmaxima in the 1D posterior probability distribution of $R_{\rm quench}$ atapproximately $0.25$ and $1.0~R_{\rm{200}}$. Analyzing four distinct solutionsin the $\tau_{\rm quench}$-$R_{\rm quench}$ parameter space, nearly all ofwhich yield quiescent fractions consistent with observational data from theGOGREEN survey, we investigate whether these solutions represent distinctquenching pathways and find that they can be separated between\textquote{starvation} and \textquote{core quenching} scenarios. The starvationpathway is characterized by quenching timescales that are roughly consistentwith the total cold gas (H$_{2}$+H{\scriptsize I}) depletion timescale atintermediate $z$, while core quenching is characterized by satellites withrelatively high line-of-sight velocities that quench on short timescales ($\sim0.25$ Gyr) after reaching the inner region of the cluster ($\lt0.30~R_{\rm{200}}$). Lastly, we break the degeneracy between these solutions bycomparing the observed properties of transition galaxies from the GOGREENsurvey. We conclude that only the \textquote{starvation} pathway is consistentwith the projected phase-space distribution and relative abundance oftransition galaxies at $z \sim 1$. However, we acknowledge that ram pressuremight contribute as a secondary quenching mechanism.

When the Well Runs Dry: Modeling Environmental Quenching of High-mass Satellites in Massive Clusters at \boldmath$z \gtrsim 1$