Exact-value conjecture for the multi-nut caching game

Consider the caching game with $k=j\ge2$, where the hider caches $k$ nuts, the searcher aims to find all $k$, and the parameters include $n$ holes and search depth $h$. The source proves the upper bound

$$\text{value}\le\frac{h^k}{{n+k-1\choose k}}.$$

Multi-nut exact-value conjecture. If

$$k=j\ge2,\qquad\frac{k+1}{k-1}\le h\le\frac nk,$$

then the value of the game is

$$\frac{h^k}{{n+k-1\choose k}}.$$

The conjecture extends the two-nut analysis; the source suggests that the integral-depth case may be easier but does not prove the full statement.