I just went over kearsarge, forester, and out shepherd over the weekend. Kearsarge is snow free. Forester has a snow field traverse on the north side that I crossed wearing approach shoes and using trekking poles, no problem. The snow was soft and I could just stomp my way through it without slowing down. If there's an unusually cold day, maybe a storm or something, it could get icy. A stumble or slip might result in a long slide into unpleasantness. That being said, it didn't seem bad at all to me and I did observe several other backpackers successfully crossing the snow in running shoes with no special gear. There's a little snow on the south side, but much less, maybe none by this weekend.