Improved lower-bound conjecture for Hilbert–Kunz multiplicities

Let $(A,\mathfrak{m}, k)$ be a formally unmixed non-regular local ring of dimension $d \geq 1$ with algebraically closed residue field. Let $m \geq 1$ be an integer, and let $A_{p,d}$ be the simple singularity defined above. Improved lower-bound conjecture. If $d=2m-1$, then either

$$\widehat{A}\cong A_{p,d}$$

or

$$\operatorname{e_{HK}}(A)>\frac{2^m}{2^m-1}.$$

If $d=2m$, then either

$$\widehat{A}\cong A_{p,d}$$

or

$$\operatorname{e_{HK}}(A)>\frac{2^m+1}{2^m}.$$

This strengthens the proposed universal lower bounds suggested by the known formulas in characteristic two. It is presented as an improvement of the preceding conjecture and remains open in general.