Large-depth value conjecture for the two-nut caching game

Assume the two-nut caching game with parameters $n$ and $h$, and let the value denote the optimal winning probability. Large-depth value conjecture. If

$$\frac{n+1}{2}\le\frac{h^2}{\lfloor h\rfloor},$$

then the value of the game is

$$\frac{\lfloor h\rfloor}{n}.$$

The source gives no proof or disproof.