Weak Target Conjecture

Let $G$ be a graph, let $D$ be a target, and let $|D|$ denote the total demand of $D$. Write $\pi(G,D)$ for the pebbling number of $G$ with target $D$, and $\pi_t(G)$ for the $t$-fold pebbling number.

Weak Target Conjecture. Every graph $G$ satisfies

$$\pi(G,D)\leq \pi_{|D|}(G)$$

for every target $D$.

The conjecture is known for trees, cycles, complete graphs, cubes, 2-paths, and Kneser graphs $K(m,2)$, according to the source, but no general proof is supplied.