Elliptic-point count conjecture for prime square-tiled surfaces in H(2)

Let $n$ be prime, and consider the elliptic points associated with primitive $n$-square-tiled surfaces in ${\mathcal{H}}(2)$. Elliptic-point count conjecture. For prime $n$, the number of elliptic points is

$$\left\lfloor\frac{n+1}{4}\right\rfloor.$$

The preceding discussion only gives an $O(n)$ upper bound from lattice points in a quarter-circle, while this formula proposes the exact count; its status is not resolved in the supplied text.