Spectra have been obtained of 21 white dwarfs (WDs) in the direction of the young, rich open star cluster NGC 2099. This represents an appreciable fraction (>30%) of the cluster's total WD population. The mean derived mass of the sample is 0.8 Msun - about 0.2 Msun larger than the mean seen among field WDs. A surprising result is that all of the NGC 2099 WDs have hydrogen-rich atmospheres (DAs) and none exhibit helium-rich ones (DBs), or any other spectral class. The number ratio in the field at the temperatures of the NGC 2099 WDs is DA/DB ~ 3.5. While the probability of seeing no DB WDs in NGC 2099 solely by chance is ~2%, if we include WDs in other open clusters of similar age it then becomes highly unlikely that the dearth of DB WDs in young open clusters is just a statistical fluctuation. We explore possible reasons for the lack of DBs in these clusters and conclude that the most promising scenario for the DA/DB number ratio discrepancy in young clusters is that hot, high-mass WDs do not develop large enough helium convection zones to allow helium to be brought to the surface and turn a hydrogen-rich WD into a helium-rich one.