Stable-process scaling conjecture for long-range percolation

Let $d\ge1$ and consider long-range percolation obtained by adding to $\mathbb Z^d$ a bond between every two distinct sites $x,y\in\mathbb Z^d$, independently, with probability proportional to $|x-y|^{-(d+\alpha)}$.

Long-range percolation scaling conjecture. If $\alpha\in(0,2)$, then the corresponding random walk scales to a symmetric $\alpha$-stable Lévy process in $\mathbb R^d$.

This extends the invariance-principle perspective from nearest-neighbor percolation to long-range percolation. The source does not provide a resolution, so the claim remains open here.