Scaling-limit conjecture for the first detachment time

Let $\hat\tau^{(n)}$ denote the first detachment time in the $n$-detachment process. Scaling-limit conjecture. The random variables

$$\frac{\hat\tau^{(n)}}{n^2}$$

have a limiting distribution as $n\to\infty$. This conjecture concerns the asymptotic behavior of the first detachment time under quadratic scaling; the source does not state whether the limiting distribution is known or identify it.