It requires only heavier snowfalls as the colder temperature is always there… colder temperatures in the more souther points of the northern hemisphere is also required to extend the range of ice.
But ask yourselves, how does the precipitation in the form of snow and ice, sleet, hail, GET into the atmosphere in larger amounts in the first place to create heavier snow and ice falls? The answer is greater heat in the tropics to evaporate water from the oceans to put that water vapor into the upper stratosphere where it migrates both north and south to convert to heavier ice crystals and precipitates and falls.
In other words, increases in greenhouse gases are MORE likely to cause a new ice age than melt the ice caps.
Decades ago, Antarctic ice cores showed that CO2 rose thousands of years after the climate warmed, meaning that the CO2 resulted from it, rather than the other way around — probably due to increased biological activity.