The extremal configuration conjecture for the $pi$-pebbling function

Let $G$ be a graph, let $p_i$ denote the pebbling demand associated with vertex $i$, and let $pi^{t}(G)=k$. A configuration assigns $c_i$ pebbles to each vertex, and it is non-$t$-solvable if it cannot be pebbled to a configuration satisfying the relevant $t$-pebbling requirement. For the standard price function, let $d$ be the diameter of $G$.

Extremal configuration conjecture. There exists a non-$t$-solvable configuration $C$ of $k-1$ pebbles on $G$ such that, for every vertex $i$, $c_i$ is either $0$ or $p_i-1$, except possibly at one vertex $r$. For the standard price function, $r$ should have another vertex $s$ at distance $d$ from it, and there should be a shortest path from $r$ to such an $s$ containing no pebbles other than those on $r$.

This conjecture proposes a generalization of the Cover Pebbling Theorem to non-positive weight functions. Establishing such a description of an extremal configuration would make it easier to calculate the $pi$-pebbling function for graphs such as wheels.