Strong Watanabe–Yoshida conjecture for Hilbert–Kunz multiplicity

Let $d\geq 2$. Set

$$R_{p,d}:=\mathbb{F}_p[[x_0,\ldots,x_d]]/(Q_d),$$

where

$$Q_d=\begin{cases}x_0x_1+x_2x_3+\dots+x_{d-1}x_d & \text{if $d$ is odd},\x_0x_1+x_2x_3+\dots+x_{d-2}x_{d-1}+x_d^2 & \text{if $d$ is even}. \end{cases}$$

Let $(R,\mathfrak{m},k)$ be an unmixed non-regular local ring of characteristic $p$ and dimension $d$. Here, unmixed means that $\dim \widehat{R}/\mathfrak{p}=\dim \widehat{R}$ for every $\mathfrak{p}\in\operatorname{Ass}(\widehat{R})$. Strong Watanabe–Yoshida conjecture. The following assertions hold:

1. $e_{\operatorname{HK}}(R)\geq e_{\operatorname{HK}}(R_{p,d})$.
2. If $e_{\operatorname{HK}}(R)=e_{\operatorname{HK}}(R_{p,d})$, then $\widehat{R}\cong R_{p,d}\widehat{\otimes}_{\mathbb{F}_p}k$; that is, the completion of $R$ is isomorphic to the coordinate ring of a quadric hypersurface.
3. $e_{\operatorname{HK}}(R_{p,d})\geq 1+c_d$, where $c_d$ are the coefficients of the Taylor expansion of $\sec x+\tan x$.

The first two assertions give a lower bound for Hilbert–Kunz multiplicity and characterize the rings attaining the minimum. The third gives a characteristic-free lower bound; it is known for sufficiently large $p$ and, according to the source, for all $p\geq 3$, while the full conjecture remains open.