Sharpness conjecture for the two-nut discrete double-limit bound

Assume the two-nut case $k=j=2$. Let the value of the game be bounded above by the quantity in the source theorem. Two-nut sharpness conjecture. The bound

$$\frac{2h^2}{n(n+1)}$$

is sharp if

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

and either $h\ge3$ or $h=3-\frac1q$ for some $q\in\mathbb Z^+\setminus\{3\}$. The source gives no proof of this conjecture.