The higher-dimensional Hilbert–Kunz lower-bound conjecture

Use the notation above: let $d\geq1$, let $p>2$ be prime, define

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

and let $(A,\mathfrak m,k)$ be a $d$-dimensional unmixed local ring with $k=\overline{\mathbb F_p}$. Higher-dimensional Hilbert–Kunz lower-bound conjecture. If $e(A)\geq3$, then

$$e_{\rm HK}(A)\geq\frac{c_d+1}{d!}.$$

The text states that this conjecture also holds when $\dim A\leq4$, while leaving the higher-dimensional cases open.