As far as I saw, it is melted out to 6000 feet most places now. The snow started below that on Lostine east fork, but not much, and that is a narrow canyon that opens to the north. I think it was unique as rangers told me many places were melted out to 7000 feet. The loop road was completely snowless at 6400, so I think you will be fine.
I called the rangers for the Elkhorns a couple weeks ago and they said you could access the route at the Anthony Lake trailhead, but the higher parts still needed a couple more weeks.