Several recent experiments on glycerol and other supercooled liquids have suggested that dynamically heterogeneous domains in the liquid are responsible for the nonexponential structural relaxation dynamics observed near the glass transition. In addition, it appears that the exchange of domain types may occur much more slowly than intradomain relaxation, at least in some cases. By examining a simple model of a dynamically heterogeneous liquid, we show that these ultraslow domain-exchange dynamics should be apparent at long times in nonequilibrium, physical aging experiments where the distribution of domains is perturbed. We recently observed such an ultraslow tail in the physical aging dynamics of the Brillouin shift for supercooled glycerol, following a rapid quench to a temperature below the glass transition (Miller, R. S.; MacPhail, R. A. J. Chem. Phys. 1997, 106, 3393); here we describe physical aging experiments carried out at two additional final temperatures. Unfortunately, the precision in the newer measurements at long times is not sufficient for us to resolve an ultraslow component. However, the broad distribution of relaxation times that characterizes the aging dynamics is consistent with the presence of such a tail and is much broader than the distribution for the equilibrium supercooled liquid. While our new experiments are ambiguous as to the nature of this broadening and the ultraslow tail, we believe the notion that the physical aging process can be sensitive to spatially heterogeneous dynamics is important, and we suggest further experiments designed to address this issue.