The first time I tried Whitney was June 19, 2011. The winter of 2010-11 had about the record for most snow (until this year maybe). I think we hit snow somewhere past Outpost Camp, around 11,000 feet. I would expect similar conditions this year around the same time.
In 2011 we had no experience in snow but rented crampons and gave it a shot anyway. Going up the chute was physically challenging, but I felt pretty comfortable otherwise. Trail crest was as far as we got on account of it being late in the day. I did find going back down the chute pretty intimidating, having no experience doing something like that.