Asymptotic optimal pebbling conjecture for eight-wide staircase graphs

Let $S_{8,n}$ be the eight-wide staircase graph, and let $\pi_{\operatorname{opt}}(G)$ be its optimal pebbling number, the minimum size of a solvable pebbling distribution on $G$.

Eight-wide staircase conjecture.

$$\pi_{\operatorname{opt}}(S_{8,n})=\frac{5}{4}n+O(1).$$

The paper obtains the value for $S_{8,8}$ by computer search but notes that even the $n=9$ case is computationally difficult. The conjecture predicts the asymptotically optimal growth rate for the eight-wide family.