Yes, early June will have high elevation trails still closed and blocked off due to snow. Went up to 9,000 feet a couple of days ago, still lots of snow. June is a very high runoff month, but I have seen in some years trails closed till 4th of july weekend.
I would call one of the parks in marysvale and ask them about the trails you plan to run and if they are aware if the trails are open or not?
plan to have hot weather down low and cool/sometimes cold weather up high and you will be fine.