On exotic Diophantine triples in $\mathbb{R}[X]$

Originally, an exotic Diophantine triple is a set $\{a,b,c\}$ of distinct nonzero rational numbers for which $$a+1,b+1,c+1,ab+1,ac+1,bc+1,abc+1$$ are all perfect squares. We prove that there is no such triple in $\mathbb{R}[X]$, with at least one nonconstant element, if none of $a,b,c$ is equal to $1$. Equivalently, under the distinct nonzero convention, every exotic Diophantine triple in $\mathbb{R}[X]$ with a nonconstant element must contain the element $1$.