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Thanks! Faithful follows my assumption on compatibility with fiber functors, so that's ok. Would be really happy to see the extension to connected reductive groups.
I see that $H^*(G/T)$ and $H^*(BT)$ are even. But in the spectral sequence $H^*(BG, H^*(G/T)) \implies H^*(BT)$, but I guess this doesn't immediately rule out $H^*(BG)$ having odd terms that cancel.
