That same phenomena has happened with at least 3 instruments, one was a 12 string which I used to tune to some DADGUM tuning(it's been so many years I can't remember which one), but at least one whole step below concert pitch, then down from there. Everytime I tried to touch the harmonics it sounded way sharp.
Then there's this funky 17 stringed not-exactly-an-oud-or-lute-whatchamacallit I got years ago from an Oud maker in Northern Turkey where they still play quarter tone music. The tuning is really funky & the harmonics always go wildly sharp. I once had the gauge of the steel strings measured by a guitar repairman pal so I could replace them. No doubt that exchanging original strings with different ones made it more pronounced, as the strings have always seemed loose.
    And, I have a Neapolitan Mandolin (rounded back with 33 ribs) which has a remarkable voice, but if I pluck one course of strings, the sound goes dramatically sharp as it sustains & fades.
    What I want to know is, why do some instruments have strings that go sharp & I have to lower the pitch to get them in tune, and some instruments are the opposite?