The asymptotic constant for maximum leaf height in the critical beta-splitting random tree

Let $D_n^*$ denote the maximum leaf height of the continuous-time critical beta-splitting random tree with $n$ leaves. Let $\mu$ and $\sigma^2$ be the constants defined in the surrounding analysis, and set

$$c:= 1 + \mu + \frac{\mu^3\sigma^2}{2} = 1.878\ldots.$$

Maximum-leaf-height constant. The constant in the asymptotic relation $D_n^* \sim c\log n$ in probability is

$$c:= 1 + \mu + \frac{\mu^3\sigma^2}{2} = 1.878\ldots.$$

This identifies the constant sought in the open problem concerning the maximum leaf height of the continuous-time model; the supplied passage gives the proposed value but does not establish the asymptotic result or provide evidence that it has been proved.