Watanabe–Yoshida's conjecture on the minimum Hilbert–Kunz multiplicity of singularities

Let $p$ be an odd prime and let

$$Q_{p,d}=\boldsymbol{F}_p[[x_1,\ldots,x_d]]/(x_1^2+\ldots+x_d^2),$$

which is a singular ring of characteristic $p$ and dimension $d-1$. For a Noetherian local ring $(R,\mathfrak{m})$ of characteristic $p$, write $e_{HK}(R)$ for its Hilbert–Kunz multiplicity, and call $R$ formally unmixed when it has the corresponding formal unmixedness property. Watanabe–Yoshida's conjecture. (1) For any formally unmixed non-regular local ring $R$ of characteristic $p$ and dimension $d$,

$$e_{HK}(R)\geq e_{HK}(Q_{p,d+1}).$$

(2)

$$e_{HK}(Q_{p,d})\geq \lim_{p\to\infty}e_{HK}(Q_{p,d}).$$

(3) For any formally unmixed non-regular local ring $R$ of characteristic $p$ and dimension $d$,

$$e_{HK}(R)=e_{HK}(Q_{p,d+1})$$

if and only if $R$ and $Q_{p,d+1}$ are isomorphic up to completion and change of base field. This conjecture seeks the exact minimum of Hilbert–Kunz multiplicity among singular local rings and describes its dependence on the characteristic. The source gives no resolution status for these assertions.