Average jump number conjecture for grids

Let $[m]^n$ denote the $n$-dimensional grid poset, and let $\bar{s}([m]^n)$ be its average jump number. The parameters $m$ and $n$ range over values for which $m^n\rightarrow\infty$. Average jump number conjecture.

$$\bar{s}([m]^n) = m^n (1-o_{m,n}(1)).$$

In other words, even when the dimension is large, the average jump number should be close to the maximum jump number. The conjecture concerns the asymptotic behaviour of random linear extensions of grid posets; the supplied source gives no resolution, so its status remains open.