Does this work for homotopy as well?

Ah, I see. They do furnish another example, but you're right that this is not the generalization I am looking for. Thanks again!

Thanks Jason! I figured the result was too special to generalize nicely to positive characteristic. I see now that I really should have thought and googled more thoroughly before posting this question on MathOverflow, since I just found this paper which claims to partially, weakly generalize Saito's aforementioned result to charateristic $p$: mathematik.uni-kl.de/~pfister/Artikel/AouiraPfister-MZ91.pdf .