@Failix thanks – I see this holds for all height-2 tricots with odd affine factors. And for 3n+1 , the members of a de Bruijn cycle v of length k=8 have parity vectors 000…, 001…, 010…, etc. Each member is valued \cfrac{\beta(v_{rot})}{2^k-3^x}, where \beta(000y_0) = 8 \beta(y_0), while \beta(001y_1) = 8 \beta(y_1) + (3^{x-1}4) , etc.
Your suggestion and the summary post here work very well for ruling out any integer cycle with k \approx 2x (where x = total odds), whether de Bruijinish or not.
I can see your idea is to push away from k \approx 2x first, then afterward exploit the low-LRS-length property of de Bruijnish cycles … I’ll keep wondering about that.