You shouldn't have a problem. We have had a significant amount of
late season snow, especially in the Mammoth area and Yosemite high
country. But the passes should be clear by late June. You may find
some patches of snow on the north side of Donohue, but it should be
mushy and easy to cross. By the time you reach Forester, it will
almost certainly be clear.
I would not carry an ice axe.
I've been using this site for years to check the snow:
For station ID, enter: UTY (this is Upper Tyndall creek, just south
Leave sensor blank, enter a start date (like 5/1/2003) and "now" for
Click on the last item, Snow Water Content (revised)
From my experience, all the passes are clear about 30 days after
this sensor reads 0.0. Right now this sensor is reading 22.5 inches
of water content.
Also, check back on this thread, I'm sure some people will be hiking
parts of the trail in June.
Here are a couple of live cams to check and watch the snow melt:
I hope that helps. Have a great hike!