Most of you were saying that the nut height needs to be the same as a fret, which of course leads to the discussion of a zero fret.
I replaced a nut on my Jack Casady bass with a Zeroglide nut.
https://goldtonemusicgroup.com/zeroglide/
With its offset tang fretwire, it is a very clever way to easily retrofit a zero fret to an instrument. For $20 or so, it might be a worthwhile experiment to see if it improves/fixes things.