Constant-distance conjecture for spatial inhomogeneous random graphs

Let $\mathbb{G}_n=G(\mathbf{Y}^{(n)},\mathbf{W}^{(n)},\kappa_n)$ satisfy the assumptions of the local weak convergence theorem in the source, except for the assumption imposing regularly varying domination on the connection function. Instead, suppose that the limiting connection function $\kappa$ satisfies that $\mathbb{E}[\kappa(t,W_0,W_1)]$ is regularly varying with exponent $\alpha\in(0,d)$. Let $U_{n,1}$ and $U_{n,2}$ be two uniformly chosen vertices in the SIRG $\mathbb{G}_n$. Constant-distances conjecture. Conditionally on the event that $U_{n,1}$ and $U_{n,2}$ are connected in $\mathbb{G}_n$,

$$d_{\mathbb{G}_n}(U_{n,1},U_{n,2})\xrightarrow{\mathbb{P}}K(\alpha,d),$$

as $n\to\infty$, where $K(\alpha,d)$ is a constant depending only on the exponent $\alpha$ and the dimension $d$. The conjecture predicts bounded typical distances in the infinite-mean degree regime $\alpha\in(0,d)$; the source does not establish the asserted convergence or identify the constant $K(\alpha,d)$.