Impossible to say at present. A La Nina winter is predicted for the Northwest, which could bring more than the usual amount of snow, meaning a southbound hike from Canada at the beginning of June would be a snowy slog. Should it be a dry winter and warm spring, a so-bo hike then would be more plausible.
As you probably know hiking through the desert in June would be miserable, if not downright dangerous. I'm pretty sure nobody's placing water caches that late.
Maybe start northbound at Kennedy Meadows in June, then do the Mexico-KM segment at the back end? Or, if you're a fast hiker, start in Canada later (July?) and hike through.
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