[MidlifeFlyer]

I'm not al all sure about this, but the reg talks about

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A graduation certificate showing that, within the preceding 3 calendar months, the person has successfully completed an approved flight instructor refresher course consisting of ground training or flight training, or a combination of both.
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So it seems to me that you must complete the course before expiration, not necessarily have the final paperwork into the FSDO by then.

But, as I said, I'm not at alll sure about this, so ask AF.