With open strings, and plucking directly over a pickup in the P position, the dominant harmonics are a 1st octave, perfect 5th and 2nd octave. But the harmonic content of a note changes depending on the LH fretting position.
It crossed my mind that it might be some sort of neck/body resonance causing the G/D string issue, but that would only affect certain notes/frequencies, it wouldn't be a general effect up and down the fingerboard.
I had something similar on my Hofner Club bass where the D string (Hofner flats) had a sort of hollow ringing sound around the 6th fret. Eventually traced it to the tailpiece which had a sympathetic vibration at that frequency. A small block of rubber carefully positioned under the tailpiece to damp the resonance fixed that.