Undecidability conjecture for rapidly growing generalised polynomials

Let $g \colon \mathbb{Z} \to \mathbb{Z}$ be a generalised polynomial such that

$$\liminf_{n \to \infty} \left|g(n)\right|/n^2 > 0.$$

Undecidability conjecture. The first-order theory of $(\mathbb{Z};<,+,g)$ is undecidable.

This conjecture proposes that adjoining any generalised polynomial with at least quadratic rate of growth to Presburger arithmetic destroys decidability, extending the known undecidability result for ordinary polynomial sequences of degree at least $2$.