Weather in the Canadian Rockies in late May, is it still snowy?

I’m considering visiting the Canadian rockies in the late spring. I’ve been there 3 times already, but all during late June when the weather’s pleasant and warm in the 60-70s…snow on the mountains but it did not snow.

I’d like to know how the weather will be in late May (last week of May). Does it still snow then? I’d like to avoid having to put chains on the car to travel from Banff to Jasper.

Please advise, thanks!