The year I was turned back was 2011, a huge snow year. The trail cleared just a couple of days later, July 12 or so, about the same time the MWMT cleared. MW cleared weeks ago, so I have no doubt the HST is clear over Kaweah Gap now. Call the SEKI/Three Rivers Wilderness office to check. and keep in mind that whatever they tell you, conditions will actually be better. CHeck this June 11 report:
High Sierra Topix Snow only on the last 100 feet below Kaweah Gap. BTW, SEKI is reporting no water at Bear Paw campground. I would assume that means at the High Sierra Camp as well, so your last water before the waterfall looks like Buck Creek. Top up there (2 liters)
