Finite truncated susceptibility conjecture for supercritical percolation

Let $G=(V,E)$ be an infinite, connected, locally finite, transitive graph, and let $K_v$ be the cluster of a vertex $v$. The truncated susceptibility is

$$\mathbf E_p\left[|K_v|\mathbbm{1}(|K_v|<\infty)\right].$$

Truncated susceptibility conjecture. This quantity is finite for every $p_c<p\leq1$.

The source describes this as a much weaker question than the preceding cluster-size-tail conjecture and states that it remains open.