Four squares from three real polynomials

We construct three nonconstant polynomials $a,b,c\in \mathbb{R}[X]$ such that $$ab+1,\qquad ac+1,\qquad bc+1,\qquad abc+1$$ are all squares in $\mathbb{R}[X]$. The entries in the construction are quadratic, so the construction has the smallest possible positive degrees. By composition with nonconstant polynomials, this gives infinitely many examples over $\mathbb{R}[X]$ with all entries nonconstant.