Conjecture on double-logarithmic diameter and typical distance in preferential attachment graphs

Let $G_{m,\delta}(t)$ be the preferential attachment graph with parameters $m$ and $\delta$, and let $V_1,V_2\in [t]$ be two uniformly chosen independent vertices. Write $\operatorname{dist}_{G}(V_1,V_2)$ for the graph distance between $V_1$ and $V_2$, and call this the typical distance. Convergence conjecture for $\delta<0$. Fix $m\geq 2$ and $\delta\in(-m,0)$. Then

$$\frac{\operatorname{diam}(G_{m,\delta}(t))}{\log\log t}$$

and

$$\frac{\operatorname{dist}_{G_{m,\delta}(t)}(V_1,V_2)}{\log\log t}$$

converge in probability to positive and different constants. The available bounds show double-logarithmic growth of the diameter in this regime, but do not determine the limiting constants or provide a matching lower bound for typical distances.