The conjecture that every graph is good

Let $G=(V,E)$ be a graph. A distribution of pebbles on $G$ is an initial arrangement of pebbles on a subset of vertices, and its support is that subset. Let $\gamma(G)$ denote the cover pebbling number of $G$. The graph $G$ is good if

$$\gamma(G)=\sum_{w\in V(G)}2^{\operatorname{dist}(w,u)}$$

for some vertex $u\in V(G)$, called a key vertex. The conjecture that every graph is good. Every graph is good. Paths, trees and complete graphs are known to be good, and the paper proves that cycles are good and that the product of any good graph with a cycle or a path is good; the general claim remains open.