Gas-rich satellites today are typically late infallers. We plot the probability distribution functions (PDFs) of the infall times of all (black), gas-poor $M_{\mathrm{gas}}^{\mathrm{sat}}<{10}^9{M}_{\odot }$ (red), and gas-rich $M_{\mathrm{gas}}^{\mathrm{sat}}>{10}^9{M}_{\odot }$ (blue) TNG-Cluster satellites at a fixed host halo $M_{200\mathrm{c}}^{\mathrm{host}}\sim {10}^{15}{M}_{\odot }$ and satellite stellar $M_{\star }^{\mathrm{sat}}\sim {10}^{11}{M}_{\odot }$ mass. We mark the medians and errors of the distributions as hashes on the top x-axis. The infall time is the first infall into any host, regardless if the galaxy has been preprocessed or not. At a fixed host halo and satellite stellar mass, the infall time is the primary driver determining whether satellites remain gas-rich or gas-poor today: more recent infallers are more likely to retain large gas reservoirs.