General graph conjecture for social connectivity time

Let $G$ be a connected graph with $n$ vertices and average degree $d$. Let $\mathrm{SC}(G)$ denote its social connectivity time. General graph conjecture. There exists a universal constant $C>0$ such that, with probability at least $1-C/n$,

$$\frac{\mathrm{SC}(G)}{(\log n)^2}\;\leqslant\; C\left(d^2\mathbf{1}_{\{G\text{ is non-regular}\}}+d\mathbf{1}_{\{G\text{ is regular but not vertex-transitive}\}}+1\right).$$

This proposes a uniform logarithmic-square upper bound, with the degree dependence distinguishing non-regular, regular non-vertex-transitive, and vertex-transitive graphs.