Asymptotic optimal pebbling conjecture for multiples of seven-wide staircase graphs

Let $S_{7k,n}$ be the staircase graph of width $7k$, and let $\pi_{\operatorname{opt}}(G)$ denote its optimal pebbling number.

Multiples-of-seven staircase conjecture. For all $k\ge 1$,

$$\pi_{\operatorname{opt}}(S_{7k,n})=kn+O(1).$$

The conjecture is motivated by duplicating an optimal distribution for $S_{7,n}$ $k$ times. It predicts the asymptotically optimal pebbling size for staircase widths that are positive multiples of seven.