Reading this makes me think that they are closing the trail indefinitely since it's a "high use trail where rescues and recoveries are common." Initially I had thought they planned to open the trail on June 30th. Should I feel confident that my July 24 permits will work out?
Yeah, you should be fine. The trail isn't closed indefinitely, it's just closed in May and June since that's when most of the rescues and recoveries occur. The vast majority of them are caused by inexperienced people who get lost or injured while travelling on snow (glissading down the chute seems to be a common theme in many of the accidents), which becomes a non-issue once the snow melts out at the end of June.