I've hiked the trail for 35 consecutive years, always in a day, but I have never gone as early as June. For maybe three of those years, I have taken an ice axe with me in the car as far as the ranger station and the trailhead. In the event that reports were really bad, I would have carried the ice axe up. However, the reports were always that hikers were making it through without any ice axe.
I carried one ski pole along one year, and since I had no ice axe with me, the ski pole was really pressed into service right before Trail Crest. Actually, any type of crampons would have been useful.
Yes, on several years the top of the switchback trail has been completely buried under snow, so hikers opted to follow the chute over closer to the face of Mount Muir. On one year, the trail was buried, and we could not work all the way over to the chute, so we simply climbed straight up the snow. Unfortunately, that caused us to hit the crest about 400 yards south of Trail Crest. That is not a catastrophic error, but it wasn't our greatest moment.
All I would suggest is that you need to take _something_ more than just boots. Maybe an ice axe. Maybe ski poles. Maybe crampons. It depends on what you are comfortable with and what the last-minute reports suggest.
The staff at the Interagency Visitor Center (permit station) should be able to tell you what to expect. However, if they size you up quickly and think you look like a beginner, they may try to scare you into bailing out. If they size you up quickly and you seem like you know what you are doing on ice, then they will not try to scare you so badly.
On a couple of years, in an attempt to lighten up my load, I went up with warm layers short of a full down jacket, and I regretted that. As long as you can keep moving, you can keep your body heat high enough to battle the wind. However, once you get up around 14,000 feet, sometimes you cannot keep moving.
Remember: More injury accidents happen on the descent.