Herscovici–Hester–Hurlbert Target Conjecture for graph pebbling

Let $G$ be a graph and let $D$ be a target distribution on $G$. Write $|D|$ for the total demand, $\pi(G,D)$ for the smallest integer $m$ such that every configuration of $m$ pebbles is $D$-solvable, and $\pi_t(G)$ for the maximum $t$-fold pebbling number over all target vertices, where $t=|D|$. Target Conjecture. Every graph $G$ satisfies

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

for every target distribution $D$. The conjecture is known for trees, cycles, complete graphs, and cubes, and the general case remains open; it is intended as a tool for studying pebbling numbers more broadly, including chordal graphs and Cartesian products.