You should check out this:
https://www.sps2022.com/post/williamson-trojan-barnard-tyndallMy guess is that you are likely to encounter snow on Shepherd's Pass, but unlikely to find much snow up from the black stain (other than maybe some in the chimney). I would think there are going to be intermittent snow patches on the North Rib. Keep in mind that the guys who did Tyndall and Williamson are (almost too) comfortable free soloing 5th class terrain.
