Observations of low-mass satellite galaxies in the nearby Universe point towards a strong dichotomy in their star-forming properties relative to systems with similar mass in the field. Specifically, satellite galaxies are preferentially gas poor and no longer forming stars, while their field counterparts are largely gas rich and actively forming stars. Much of the recent work to understand this dichotomy has been statistical in nature, determining not just that environmental processes are most likely responsible for quenching these low-mass systems but also that they must operate very quickly after infall onto the host system, with quenching timescales $lesssim 2~ {rm Gyr}$ at ${M}_{star} lesssim 10^{8}~{rm M}_{odot}$. This work utilizes the newly-available $Gaia$ DR2 proper motion measurements along with the Phat ELVIS suite of high-resolution, cosmological, zoom-in simulations to study low-mass satellite quenching around the Milky Way on an object-by-object basis. We derive constraints on the infall times for $37$ of the known low-mass satellite galaxies of the Milky Way, finding that $gtrsim~70%$ of the `classical satellites of the Milky Way are consistent with the very short quenching timescales inferred from the total population in previous works. The remaining classical Milky Way satellites have quenching timescales noticeably longer, with $tau_{rm quench} sim 6 - 8~{rm Gyr}$, highlighting how detailed orbital modeling is likely necessary to understand the specifics of environmental quenching for individual satellite galaxies. Additionally, we find that the $6$ ultra-faint dwarf galaxies with publicly available $HST$-based star-formation histories are all consistent with having their star formation shut down prior to infall onto the Milky Way -- which, combined with their very early quenching times, strongly favors quenching driven by reionization.
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