Strong-geometry conjecture for geometric preferential attachment

For $\gamma>1$, define

$$F_\gamma(r)=\exp\{(\log(1/r))^\gamma\},$$

where $\log x=\max\{0,\log x\}$. Let $G_n$ be the geometric preferential attachment graph under the paper's standing assumptions, let $v_n$ be the vertex selected by the new vertex, and let $\eta_1(n)$ be its on-line nearest neighbour. Strong-geometry conjecture. The conclusion of Theorem 1(ii), including convergence to the on-line nearest-neighbour degree sequence, is valid for every $\gamma>1$. The theorem establishes this conclusion for $\gamma>3/2$; extending it to the full range $\gamma>1$ remains open, as does the stronger almost-sure convergence discussed in the surrounding remark.