Bounded arithmetic progressions in ranges of rational polynomials

Let $g(x)$ be a polynomial over the rationals, and write $g(\mathbb{Q})$ for its range on rational inputs. An arithmetic progression is a finite sequence of the form $a+id$ with fixed rational numbers $a,d$ and consecutive integer indices $i$. Arithmetic-progression conjecture. For every rational polynomial $g(x)$, there is a bound $b$ such that $g(\mathbb{Q})$ contains no arithmetic progression of length $b$.

This would rule out arbitrarily long arithmetic progressions in polynomial ranges and, together with the preceding additive-combinatorial proposition, would constrain rational polynomials that fail to expand. The authors state that they were unable to prove it and that it would follow from the Uniformity Conjecture.