I can speculate... of course you can just ask when you get your permit.
I, for one, never go back and report conditions though.
BV Lake will prob. be still mostly frozen over. Snow up there. No snowshoes required anymore.
Snow Level prob. about 9500. Getting over Chil. Creek may be quite an issue.
Both Horseridge and BV Crests ... the north facing... were holding snow every week I saw them.
Last year mid-June pretty much no snow below Edson. Expect that now.
With GP road open, why though. Just go up Ostrander and x up.
Unless of course you want to see Chil. Falls.
Looked like this in early April:
http://yosemitenews.info/forum/read.php?3,52472Never seen it as pretty as that.
Sorry can't help more. I do that stuff when GP road and Tioga are closed
Have fun