Uniformly bounded geometric progressions in translated integer polynomial ranges

Let $g$ be a polynomial over the integers. For each integer $a$, define

$$h_a(x)=g(x+a)-g(a).$$

A geometric progression of length $b$ is a sequence of $b$ terms with a common ratio. Geometric-progression conjecture. There is a bound $b$ such that, for every $a\in\mathbb{Z}$, the range $h_a(\mathbb{Z})$ contains no geometric progression of length $b$; equivalently, $GP_b\notin h_a(\mathbb{Z})$.

The source presents this as a weak form of a uniformity conjecture that would imply the additive statement in the integer case. The authors say they were unable to prove even the corresponding assertion that an integer polynomial of degree at least two has no arbitrarily long arithmetic progressions in its integer range.