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 \[ a+1,\quad b+1,\quad c+1,\quad ab+1,\quad ac+1,\quad bc+1,\quad abc+1 \] 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 $1$.