Sharpness conjecture for the discrete-limit upper bound

Assume the two-nut caching game and let the discrete-limit theorem provide its stated upper bound for $h$. Discrete-limit sharpness conjecture. If

$$h\in\left(\frac52,\frac83\right)\cup\left[\frac{19}{7},2\right)\setminus\left\{3-\frac1q:q\in\mathbb Z^+\right\},$$

then the best upper bound supplied by that theorem is sharp. The source does not establish this claim.