Sharpness conjecture for the discrete double-limit caching game

Let $v(k,j,n,h)$ be the value of the discrete caching game and let $v^*(k,j,h)$ be the corresponding limit-game value. The source states the upper bound

$$v(k,j,n,h)\le {n+j-1\choose j}v^*(k,j,h).$$

For $k=j=2$, the transformed hiding strategy is obtained from the limit game by assigning the corresponding depths to the discrete holes. Sharpness conjecture. For $k=j=2$ and every $h$, when $n$ is sufficiently large, the bound is sharp and the transformed hiding strategy in the limit game is optimal. This is an open conjecture in the source.