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Indeed, at $\mstar\sim10^9\msun$, galaxies transition through $\mgas/\mstar\approx1$, with gas depletion timescales comparable to a Hubble time.  In this scenario, the quenching timescales at $\mstar\gtrsim10^9\msun$ do not necessarily require strong additional environmental processes other than the lack of gas accretion to account for quenching in satellites \citep[see related discussions in, e.g.,][]{Wetzel2013,Wheeler2014,McGee2014}.  However, this scenario cannot explain the rollover in the quenching time at $\mstar\lessim10^9\msun$, $\mstar\lesssim10^9\msun$,  because the star-forming dwarf galaxies of the LG also have $\mgas\gtrsim\mstar$ \citep{GrgvichPutman2009}, and thus contain enough gas to fuel star formation for a Hubble time, even absent accretion. Thus, the rapid decline of the environmental quenching time \emph{requires} an additional process that can remove gas from these dwarf galaxies after infall.  This likely arises from ram-pressure stripping of cold gas in such satellites, whose lower-mass host (sub)halos have shallower potential wells.  Furthermore, the same internal stellar feedback that regulates the low star-formation efficiency in such dwarf galaxies and likely drives significant gas flows to large radii \citep[e.g.,][]{Muratov2015} would strongly assist such environmental stripping to make even more efficient in dwarf galaxies.