SP Trail is predominantly south facing and for the exception of the pass headwall not exposed. So I imagine the snow should be very melted out pending further storms.
I think that conditions past the pass will be marginal to miserable.
The way past the Pass is sandy to the Tyndall-Polychrome Col, shouldn't require more than micro-spikes up until this point.
Though I am more than open to correction, at this time of year there will be light snow cover in various reaches of the Williamson Bowl, and especially in the West Chute and the Chimney. Since the summer conditions are endless scralus, and generally cruddy, and due to the thin snow cover it won't be consolidated enough to justify crampons/axe but it will be deep enough to be dangerous and slippery.
You'll also be a long way from Kansas up there too

I'd wait a month or two until there is more snow, and some bona fide consolidated pack.
But that's just me.