Benjamini's conjecture on the asymptotic Cheeger constant of the giant percolation component

Let $d\geq 2$ and $p>p_c(d)$. For the giant component $\textbf{C}_n$ of supercritical bond percolation in $[-n,n]^d$, let $\Phi_{\textbf{C}_n}$ denote its Cheeger constant. Benjamini's conjecture. The limit

$$\lim_{n\to\infty} n\Phi_{\textbf{C}_n}$$

exists $\mathbb{P}_p$-almost surely as a deterministic constant in $(0,\infty)$. This conjecture concerns the precise asymptotic constant behind the known order $\Phi_{\textbf{C}_n}\asymp n^{-1}$; the supplied status evidence indicates that the associated limit shape is known as the Wulff shape $W_p$ and solves the corresponding deterministic continuum isoperimetric problem.