Can the polynomial $$y^2(1-x)^4+x^2(1-y)^4=xy(1-x)^2(1-y)^2(n^2+2)$$ be solved over the roots of unity for an integer $n\ge 3$? Known solutions are $$n=1:\quad x=\zeta_5\quad y=\zeta_5^2$$ $$n=2:\quad x=\zeta_8\quad y=\zeta_8^3$$
As the problem is posed, $x$ and $y$ do not have to necessarily be roots of unity of the same degree. But as far as I know, that may be a necessary condition for solvability.
This is a reformulation of an earlier question Rational Trig Solutions for $n\ge 3$ I was advised to formulate the question as it is presented here because there are apparently strong results on what polynomials can be solved over the roots of unity. There are a few papers on such solutions that pop up on a naif google search, but all have been too dense for me to understand.