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TPMS for full-size spare tire?

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Due to having accident damage repaired, I find myself with a extra rim that's in serviceable condition. Since I also kept a tire from the last time I bought new rubber, I'm close to having a full size spare for the occasional trip off the beaten path.
The question is whether I should obtain another TPMS sensor for the spare, or if I should just put in a plain old valve stem. Is there a possibility that having a spare in the trunk with a working TPMS sensor will confuse the car? Or is this probably far enough from the receiver in the wheel well that I shouldn't have to worry about it?