Question about snowfall in Glacier National Park?

I am planning on driving through Glacier National Park during my road trip this summer. Has there been more than average snowfall there this winter. If so, if I'm going through there in mid-late June, will I run into any serious snow (say more than a few inches)? And can I expect to see snow anywhere else in the Pacific Northwest? If there is expected to be more snow this year, then would that push the snow clearing back a few weeks (when do they normally clear off Going to the Sun road in Glacier NP)? Do they expect to be delayed this year if there is more snow? Would I be better off driving through in the first week of July? Any advice would be greatly appreciated.