Linear chemical distances in scale-free Gilbert graphs

Let $G(X^{(n)})$ be the scale-free Gilbert graph on the point set $X^{(n)}$ in the torus $\mathbb{T}_n$, let $\mathsf{e}_1=(1,0,\ldots,0)$, and let $q(x)$ denote the closest point of $X$ to $x$. Assume $s>d$. Linear chemical-distance conjecture. There exists a constant $c=c(\beta,d)>0$ such that the chemical distance between $q(-n\mathsf{e}_1/4)$ and $q(n\mathsf{e}_1/4)$ is at least $cn$ with high probability as $n\to\infty$. The preceding theorem proves only a lower bound of order $n/(\log n)^{\alpha}$ for every $\alpha>0$; the conjecture asserts that this sublogarithmic loss is an artifact of the proof and that distances are instead bounded below linearly.