The non-definability conjecture for asymptotic couples in precontraction groups

Let $(\Gamma,\chi)$ be a nontrivial precontraction group, where $\Gamma$ is an ordered abelian group and $\chi:\Gamma^{\neq}\to\Gamma$ is its precontraction map. A function $\psi:\Gamma^{\neq}\to\Gamma$ is definable if it can be defined in $(\Gamma,\chi)$, possibly using parameters. The notation $\Gamma^{<}$ denotes the negative elements of $\Gamma$.

Non-definability conjecture. In no nontrivial precontraction group $(\Gamma,\chi)$ can one define, even allowing parameters, a function $\psi:\Gamma^{\neq}\to\Gamma$ such that $(\Gamma,\psi)$ is an $H$-asymptotic couple and

$$\chi+\psi\circ\chi=\psi$$

on $\Gamma^{<}$.

This conjecture asserts that the precontraction structure cannot definably recover an $H$-asymptotic-couple map satisfying the displayed compatibility with $\chi$. The paper presents it as an unresolved issue arising from the use of shifts and does not provide a resolution.