The absence of infinitesimals above 1 in maximal Hardy fields

Let $H$ be a maximal Hardy field, let $H^{>\mathbb R}$ denote the elements of $H$ greater than every real constant, and write $f\prec g$ when $f/g\to 0$ at $+\infty$. A function $y\in\mathcal{C}^1$ is required to have derivative $y'\in\mathcal{C}^\times$, the nonvanishing $\mathcal{C}^1$ germs.

Maximal Hardy-field conjecture. If $H$ is maximal, then there is no $y\in\mathcal{C}^1$ such that

$$1\prec y\prec h\quad\text{for all }h\in H^{>\mathbb R},$$

and $y'\in\mathcal{C}^\times$.

This conjecture is presented as much stronger than the preceding proposition, postulating an analogue of the cited lower-bound result for infinite lower bounds. Its resolution is not given in the supplied text.