The critical-exponent conjecture for long-range percolation cluster diameter

Let $N$ be a positive integer, let $s,\beta>0$, and let $G_{s,\beta}(N)$ be the random graph on the cycle $\mathbb{Z}/N\mathbb{Z}$ in which cycle-neighboring vertices are joined and distinct vertices at cyclic distance $r>1$ are joined independently with probability $1-\exp(-\beta r^{-s})$. Let ${\cal D}(N)={\cal D}_{s,\beta}(N)$ denote its diameter.

Critical-exponent conjecture. (A) If $s=2$, then the diameter's order of magnitude is $N^\delta$, where $\delta$ is a function of $\beta$. (B) If $1<s<2$, then the diameter is $\theta(\log(N)^\gamma)$, where $\gamma>1$ is a function of $s$.

The source records the $s=1$ part of the earlier conjecture as contradicted by a result giving order $\log(N)/\log\log(N)$, and says that the $1<s<2$ part was proved by Biskup. The $s=2$ assertion remains open in the supplied text.