If you can correctly intonate it with a capo on the first fret, it strongly suggests that the nut is at fault.
As an aside, I wouldn't bother with a strobe tuner. The purpose of intonation is so it plays in tune no matter what fret is pressed, and a regular tuner will tell you if it is or not. OTOH, I have a clip on snark tuner that doesn't seem terribly accurate, I just get it close with that and use relative tuning to get the strings in tune with each other. There's a windows tuner called that seems to work better than any hardware tuner I've ever had, even the old ones that you needed to plug a cable into.
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