Long term and short term fuel trim are all dictated by the first oxygen sensor. The second sensor is just there to verify that the catalytic converter is working properly.
Just because your short term trim is positive while idling doesn't mean it'll be positive when accelerating or cruising. There are tons of variables that control how much fuel is injected. Unless your LTFT or STFT are being maxed out, I would doubt there is an issue. If your LTFT is ALWAYS negative, and STFT is ALWAYS positive, then you might have an issue. I don't see how this could happen as its all just programming in the ECU. I don't see how a defective sensor could cause this to happen.
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