Constancy conjecture for the long-range percolation asymptotic function

The model is long-range percolation on $\mathbb R^d$, with chemical distance $D(0,rx)$ and asymptotic function $\mathit\phi:(1,\infty)\to(0,\infty)$ arising in the sharp distance asymptotic; $\beta>0$ denotes the intensity parameter.

Constancy conjecture. The function $\mathit\phi$ above is constant for each $\beta>0$.

The function $\mathit\phi$ encodes the dependence of the limiting asymptotic on $\beta$ and the underlying norm. The authors explain that it may be an artifact of their subadditivity argument and state that they do not know how to prove the conjecture.