We make this trip several times a month, and we have never had an occasion where they required us to put chains on a 4WD - if the conditions are that bad, they will close the road.
However, the vast majority of the time, they just check that your SUV has 4WD and they wave you through, while everyone else is putting on chains.
Also - most rental car companies do not permit you to use chains, due to the potential for damage.