Integral-depth multi-nut exact-value conjecture

Consider the caching game with $k=j\ge2$, in which the hider caches $k$ nuts and the searcher aims to find all of them. Integral-depth multi-nut conjecture. If

$$k=j\ge2,\qquad h\in\mathbb Z^+,\qquad h\le\frac nk,$$

then the value of the game is

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

The source presents this as a weaker, seemingly easier form of the preceding conjecture and gives no proof.