Supercritical case of the upper-bound conjecture for chemical distance

Let $D(0,x)$ denote the chemical distance between $0$ and $x$, with $D(0,x)=\infty$ when they are disconnected, and let $1_{\{0\leftrightarrow x\}}$ indicate that they are connected. Assume the hypotheses of the infinite-cluster upper-bound conjecture, including that almost surely there exists an infinite cluster, and additionally assume that the system is supercritical. Almost surely,

$$\limsup_{\|x\|\to\infty}\frac{1_{\{0\leftrightarrow x\}}D(0,x)}{\|x\|}<\infty.$$

Supercritical upper-bound conjecture. The displayed limsup is finite.

The source presents this as a special case of the infinite-cluster case, but gives no resolution of it.