If z is a Gaussian integer and n > 3, when can z^n±z be a rational integer?

2022-04-30 I came across the following curious conjecture. If you have an idea how to approach it, let me know!

Conjecture. For any integer and any Gaussian integer that is not a unit, if or is a rational integer, then is a rational integer.

2022-06-07 This turns out to be true, see this MathOverflow discussion: https://mathoverflow.net/questions/421439/given-that-n-3-and-z-is-a-gaussian-integer-when-can-zn-pm-z-be-a-rati/421972#421972.