Neutral molecular densities were measured just outside the plasma edge at 14 axial locations in the Tandem Mirror Experiment Upgrade (TMX-U). Bayard-Alpert gauges can be operated routinely as calibrated pressure gauges in magnetic fields between 0.3 and 0.7 T. The pressures at the outer turning point and at the midplane of the plug are moderately low, 1 × 10−6 torr, but during plasma operation the inner turning point pressure is as high as 2 × 10−5 torr. This high pressure is consistent with a wall reflux coefficient of approximately 14, assuming that local anomalous radial loss is the cause. Fokker-Planck modelling of the plasma densities with molecular and energetic atomic fuelling shows that this high inner turning point pressure does not account for the measured plasma densities. However, an as yet undetermined additional radial transport of sloshing ions with a 1 ms rate does produce agreement. Both measurements and modelling show that the charge exchange rate at the exposed sloshing ion turning points is high (0.5 ms), but the plasma is self-shielding for most of the remaining axial distance, and the bounce averaged, charge exchange radial loss time constant is 5 ms.