Perfect cuboids and irreducible polynomials

The problem of constructing a perfect cuboid is related to a certain class of univariate polynomials with three integer parameters $a,b$, and $u$. Their irreducibility over the ring of integers under certain restrictions for $a,b$, and $u$ would mean the non-existence of perfect cuboids. This irreducibility is conjectured and then verified numerically for approximately 10 000 instances of $a,b$, and $u$.