Exponential radius-tail conjecture for finite supercritical clusters

Let $G=(V,E)$ be an infinite, connected, locally finite, transitive graph, let $K_v$ be the cluster of $v$, and let $B(v,n)$ be the graph-distance ball of radius $n$ around $v$. Exponential radius-tail conjecture. For every $p_c<p<1$ there exists a positive constant $c_p$ such that

$$\mathbf P_p\bigl(K_v\leftrightarrow\partial B(v,n),\ |K_v|<\infty\bigr)\leq e^{-c_p n}$$

for every $n\geq1$.

The source expects this exponential tail on every transitive graph, notes that the nonamenable case follows from its main theorem, and cites the Euclidean lattice case as known; the general claim remains open. Similar bounds are expected for the intrinsic radius.