Degree-distribution convergence to BB(m) in regime R2

Let $m\in\mathbb{N}$, let $G_t$ be a graph in Regime $R2$, and let $\mathbf{\deg}_t$ be its empirical degree distribution. Let $\mathbf{\deg}_t^{BB(m)}$ be the empirical degree distribution of a $BB(m)$ model with the same parameter $m$. Degree-distribution convergence conjecture in $R2$. Almost surely in the fitness realisation of $G_t$,

$$\lim_{t\to\infty}d_{TV}(\mathbf{\deg}_t,\mathbf{\deg}_t^{BB(m)})=0.$$

This proposes that the degree distribution in Regime $R2$ has the $BB(m)$ benchmark as its asymptotic limit in total variation. The claim is supported numerically and remains unproved in the supplied text.