Conjecture on the asymptotic expansion of the Bonse threshold parameters

Let $A_m$ be the sequence of threshold parameters associated with the step function $\Psi(x)$, and suppose that

$$A_m=-2+O\!\left(\frac{\log\log m}{\log m}\right)\quad\text{as }m\to\infty.$$

Asymptotic expansion conjecture. There exists a constant $\kappa>0$ such that

$$A_m=-2+\kappa\,\frac{\log\log m}{\log m}+o\!\left(\frac{\log\log m}{\log m}\right).$$

This conjecture sharpens the paper’s asymptotic estimate for $A_m$ and predicts the leading correction term; the supplied text gives no evidence that it has been resolved.