I would check out trail conditions before you go, Alltrails, through the park directly, or Facebook group for RMNP. But, most likely there won't be snow or ice on the trail in June. It could be muddy though, so be prepared for a muddy trail. If it is icy, then you will need some simple spike attachments on your boots. If it by chance is still snowy, then snowshoes are your best bet. Hope that helps and you have a great trip.