The half-minimum-degree conjecture for restricted optimal pebbling

Let $G$ be a graph, with vertex set $V(G)$, minimum degree $\delta(G)$, and pebbling numbers $\pi^*(G)$ and $\pi_2^*(G)$ denoting its optimal and $2$-restricted optimal pebbling numbers, respectively.

Half-minimum-degree conjecture. If

$$\delta(G)\geq \frac{1}{2}|V(G)|,$$

then

$$\pi_2^*(G)=\pi^*(G).$$

Graphs whose minimum degree is at least half their order have diameter at most two, so the conjecture reduces to proving the existence of a solvable $2$-restricted pebble distribution of size $4$ for every such graph. The source states that this threshold was not determined and presents this claim as the next conjecture.