I've hiked that area south of Tuolumne Meadows and past Vogelsang, and I would never attempt to go as early as June 1. It may be a little snowy at July 1.
The high point would be Vogelsang Pass, and I have hit snow there before at almost August 1. By that late, the remaining snow is tracked up. Actually, past Vogelsang Pass you are on the Lewis Creek Trail, and it does not normally get any early season trail maintenance. In fact, get in there and break your leg, and you may not be found for a week. Also, NPS budget cutbacks may limit trail maintenance to a minimum. That trail is one way to cover the High Sierra Camps, so that is the only thing that may open it earlier.
Right now we are still in mid-winter, and it is difficult to predict what the full effect of winter snowpack will be. At present, it is running around 120% of normal. That means that persisting snow drifts will be there in July.