The consensus-time conjecture for evolving scale-free networks

Consider the evolving scale-free network and the voter model in the same setup as Theorem, with arbitrary update rate $\kappa$. Let $T_{\text{cons}}$ denote the consensus time and let $\mathbb{E}_{\mu_u}$ denote expectation for the initial distribution $\mu_u$. For parameters $\beta$ and $\gamma$, Consensus-time conjecture.

$$\mathbb{E}_{\mu_u}\left(T_{\operatorname{cons}}\right)= \begin{cases} \Theta\left(N+\frac{N}{\kappa}\right), & \beta+2\gamma<1,\\[5pt] \Theta\left(N+\frac{1}{\kappa}\log N\right), & \beta+2\gamma>1. \end{cases}$$

This conjecture removes technical assumptions and polylogarithmic corrections from the preceding theorem. It predicts distinct consensus-time scales in the subcritical and supercritical regimes, but the source provides no resolution for arbitrary $\kappa$.