In the 1480s, da Vinci invented the first hygrometer using cellulose fibres to attract moisture from the atmosphere. Five hundred years later, Williams and Blanc showed that the depressed vapour pressure of a hygroscopic sessile droplet can inhibit condensation within an annular dry zone on the surface. What remains unresolved to this day is whether these regions of suppressed condensation around hygroscopic agents are due to inhibited nucleation versus inhibited growth of the condensate. We elucidate the competition between these two mechanisms by generating steady-state dry zones about frozen water droplets. The choice of ice as the hygroscopic material was motivated by its unique ability to remain undiluted as it attracts moisture from the air. Experiments, scaling models, and simulations where the ice droplet size, ambient humidity and surface temperature are systematically varied reveal that over the vast majority of the parameter space, the inhibited growth dry zone wins the duel over the nucleation dry zone.