Maximal F-signature conjecture for non-regular local rings

Let $(A,\mathfrak{m})$ be a non-regular local ring of dimension $d \geq 1$, and let $A_{p,d}$ be the simple singularity defined above. Maximal F-signature conjecture.

$$\operatorname{s}(A)\leq 2-\operatorname{e_{HK}}(A_{p,d})=\operatorname{s}(A_{p,d}).$$

Since $A_{p,d}$ is a hypersurface of multiplicity two and satisfies $\operatorname{e_{HK}}(A)=2-\operatorname{s}(A)$, this asserts that the simple singularity has maximal F-signature, equivalently minimal Hilbert–Kunz multiplicity, among the non-regular local rings under consideration. The source presents this as a natural conjecture; no general resolution is supplied.