High-dimensional critical threshold conjecture for GN_k continuum percolation

Let $\alpha_c^{(k)}(d)$ denote the critical value of the parameter $\alpha$ for percolation in the GN$_k(d,\alpha)$ model in dimension $d$, with integer $k\geq 1$. High-dimensional threshold conjecture. For all $k\geq 1$,

$$\alpha_c^{(k)}(d)\to 1\quad\text{as }d\to\infty.$$

The result is known for $k=2$; the conjecture extends the stated high-dimensional limit to every fixed connection rank $k$.