@WilliamBalderrama Thanks! My proof uses the computation of $π_*KU^{tC_2}$ (which is easy because KU is complex oriented) and the Wood fiber sequence, but this needs to invert 2 (since I want to kill $\eta$), although that's not a problem for the application I have in mind.. It's always good to have an independent confirmation!

Crazy question. Suppose I have a stable ∞-category C. Is there a way of finding a stable category D and a functor D→C such that K(D)=t_{\ge n}K(C)? For n=1 this is easy (just take the subcategory of objects with trivial K-class), but what about higher n?