The Hilbert–Kunz multiplicity lower-bound and rigidity conjecture

Let $k$ be a field of characteristic $p\geq 3$, let $d\geq 2$, and set

$$A_{p,d}=k[[X_0,X_1,\ldots,X_d]]/(X_0^2+X_1^2+\cdots+X_d^2).$$

For an arbitrary $d$-dimensional unmixed local ring $(A,\mathfrak{m})$ of characteristic $p$ with residue field $k$, Hilbert–Kunz lower-bound conjecture. If $A$ is not regular, then $e_{\rm HK}(A)\geq e_{\rm HK}(A_{p,d})$; moreover, if equality holds and $k$ is algebraically closed, then the $\mathfrak{m}$-adic completion $\widehat{A}$ is isomorphic to $A_{p,d}$. The surrounding text presents this as a natural question about a sharp lower bound, and the candidate's resolution status is not supplied.