Four-square polynomial triples: quadratic classification and equal-degree obstructions

We study triples of distinct nonconstant polynomials $a,b,c$ over $\mathbb{R}$, $\mathbb{Q}$ or $\mathbb{Z}$ for which $$ ab+1,\qquad ac+1,\qquad bc+1,\qquad abc+1 $$ are all polynomial squares. We show that, up to permutation and a common invertible affine change of variable, there is exactly one