Watanabe–Yoshida conjecture on Hilbert–Kunz multiplicities of simple singularities

Let $p$ be a prime number and define

$$\mathcal{A}_{p,n}:=\frac{\mathbb{F}_p[[x_0,\dots,x_n]]}{(x_1^2+\cdots+x_n^2)}.$$

Let $E_n$ denote the Euler number, equivalently the number of linear extensions of the zigzag poset of size $n$. Watanabe–Yoshida conjecture. The Hilbert–Kunz multiplicity satisfies

$$e_{HK}(\mathcal{A}_{p,n})\geq 1+\frac{E_n}{n!}.$$

The conjecture gives a uniform lower bound for the Hilbert–Kunz multiplicities of these simple $(A_1)$-singularities and connects the problem with Euler numbers and Ehrhart theory. The source notes that Meng announced a proof using different analytic methods, while the Ehrhart-theoretic proof is the subject of the paper.