Random-matrix conjecture for the squarefree-number detection exponent

Let $\eta_\infty(\theta)$ be the limiting exponent governing the algorithm's running time, and let $\theta^*$ be the threshold parameter defined by the preceding complexity analysis. The paper models $\eta_\infty(\theta)$ using a suitable random matrix model.

Random-matrix conjecture. The limiting exponent is

$$\eta_\infty(\theta)=\begin{cases} 1-\frac{\theta}2&\text{if }0<\theta<1,\\ \frac12&\text{if }\theta\ge1. \end{cases}$$

In particular, $\theta^*=\frac23$.

This conjectural value would imply that the algorithm can certify that $N$ is squarefree in time $O\bigl(\exp[(\log N)^{2/3+o(1)}]\bigr)$. The claim is based on an analogous calculation for a random matrix model; the source provides no resolution of the conjecture.