The linear t-pebbling bound conjecture

Let $G$ be a graph, let $t\geq 1$, and let $\operatorname{diam}(G)$ denote its diameter.

Linear t-pebbling bound conjecture.

$$\pi_t(G) \leq \pi(G) + 2^{\operatorname{diam}(G)}(t-1).$$

The source states that this is weaker than the diameter increment conjecture and proves it for graphs of diameter $2$, but gives no general resolution.