Hi,
Our trip on the HST last summer at the exact time you are planning was dry as a bone. In fact, we moved our trip up to early July after talking with folks who warned that some of the creeks could run dry in August. The only snow we saw was a few light patches on the pass way above Hamilton lake. Here is a link to our trip:
High Sierra Trip ReportI've read trip postings from folks who've done the hike in heavy snow years at the same time as you are planning and their big problem seemed to be river crossings. Some of the streams up in the Chagoopa Plateau were quite high and made for exiting passage. Here is a link to their trip:
Jason's High Sierra Trip ReportThe other place I've seen deep snow at that time of year is on the switchback down from Trail Ridge. As your trip draws close you can decide weather trekking poles will be enough or if an ice axe might be in order. If you have any doubt, bring the axe because trekking poles are ineffective when a quick self-arrest is called for. I have to say the HST is a really fabulous hike. We thoroughly enjoyed it and would do it again this summer if there weren't so many other great trips in the area.
Enjoy,
Ben