Balister–Bollobás–Sarkar–Walters sharpness conjecture for k-nearest-neighbour graphs

Let $S_n=[0,\sqrt{n}]\times[0,\sqrt{n}]\subset\mathbb{R}^2$, place points in $S_n$ according to a Poisson process of intensity $1$, and join each point to its $k$ nearest neighbours to obtain the random geometric graph $S_{n,k}$. Balister–Bollobás–Sarkar–Walters' sharpness conjecture. For any $0<\varepsilon<1$, there exists an integer constant $C(\varepsilon)$ such that for all $n$ sufficiently large, if

$$\mathbb{P}(S_{n,k}\text{ is connected})\geq\varepsilon,$$

then

$$\mathbb{P}(S_{n,k+C(\varepsilon)}\text{ is connected})>1-\varepsilon.$$

The conjecture asserts that the connectivity transition has a uniformly bounded-width sharp threshold in $k$. The paper states that it proves this conjecture.