that this process is *tail trivial*, i.e., all events in the tail sigma-algebra $${\mathcal{T}}= \cap_{N \in {\mathbb{N}}}\, \sigma\left(\cup_{n \geq N} {\mathrm{CM}}_n\right)$$ have probablity 0 or 1. Consequently, the associated communcation height process $(\Gamma^\star_n)_{n\in{\mathbb{N}}}$ with $\Gamma^\star_n = \Gamma^\star({\mathrm{CM}}_n)$ is tail trivial as well. In particular, both $\gamma^*_- = \liminf_{n\to\infty} n^{-1}
\Gamma^\star_n$ and $\gamma^*_+ = \limsup_{n\to\infty} n^{-1} \Gamma^\star_n$ exists and are constant a.s. Theorems \[thm:Ubound\]–\[thm:Lbound\] show that $0<\gamma^*_- \leq \gamma^*_+<\infty$. Settling Conjecture \[conj:scalnucltime\] amounts to showing that $\gamma^*_-=\gamma^*_+$.

[**4.**]{} It was shown by Dommers [@Dpr] that for the Configuration Model with $f=\delta_r$, $r \in {\mathbb{N}}\backslash\{1,2\}$, i.e., for a random regular graph with degree $r$, there exist constants $0<\gamma_-^\star(r)<\gamma_+^\star(r)<\infty$ such that $$\label{eq:sandwich}
\lim_{n\to\infty} \lim_{\beta\to\infty}
{\mathbb{E}}_n\left( P^{{\mathrm{CM}}_n}_\boxminus\left( e^{\beta n\gamma_-^\star(r)}
\leq \tau_\boxplus \leq e^{\beta n\gamma_+^\star(r)}\right) \right) = 1,$$ provided $\frac{h}{J} \in (0,C_0\sqrt{r})$ for some constant $C_0 \in (0,\infty)$ that is small enough. Moreover, there exist constants $C_1 \in (0,\tfrac14\sqrt{3})$ and $C_2 \in
(0,\infty)$ (depending on $C_0$) such that $$\gamma_-^\star(r) \geq \tfrac14 Jr - C_1J\sqrt{r}, \qquad
\gamma_+^\star(r) \leq \tfrac14 Jr + C_2J\sqrt{r}, \qquad r \in {\mathbb{N}}\backslash\{1,2\}.$$ The result in is derived without hypothesis (H), but it is shown that hypothesis (H) holds as soon as $r \geq 6$.

[**Outline.**]{} The rest of the paper is organised 