Sharp large-time behavior conjecture for degree evolution

Consider the Barabási–Albert model with communities. Fix a vertex $\tau$ that is not isolated in the initial graph, let $Y_T$ be its degree at time $T$, and let $\theta^*_{\ell_\tau}$ be the rate parameter associated with its community.

Sharp large-time behavior conjecture. For $\tau$ fixed,

$$\lim_{T\to\infty}\frac{Y_T}{(T/\tau)^{\theta^*_{\ell_\tau}}}=W\quad\text{almost surely},$$

where $W$ is a random variable satisfying $\mathbb{P}\{W>0\}=1$.

The preceding proposition establishes only the logarithmic growth rate, so this conjecture asks for a positive almost-sure limiting multiplicative constant and would give a sharper description of the degree evolution.