Upper semi-continuity conjecture for Hilbert–Kunz multiplicity

Let $R$ be a locally equidimensional ring of characteristic $p>0$. For each prime ideal $\mathfrak p\in\operatorname{Spec}R$, write $\operatorname{e}_{HK}(\mathfrak p)=\operatorname{e}_{HK}(R_{\mathfrak p})$, where $\operatorname{e}_{HK}(R_{\mathfrak p})$ is the Hilbert–Kunz multiplicity of the local ring $R_{\mathfrak p}$. A function on $\operatorname{Spec}R$ is upper semi-continuous if, for every $a\in\mathbb R$, the set of points where its value is less than $a$ is open.

Upper semi-continuity conjecture. If $R$ is an excellent ring, then the function

$$\mathfrak p\longmapsto \operatorname{e}_{HK}(\mathfrak p)$$

is upper semi-continuous on $\operatorname{Spec}R$. More generally, if $R$ is F-finite, then the same conclusion holds.

This conjecture asks whether the upper semi-continuity known for the finite-level Hilbert–Kunz functions extends to their limiting multiplicities. The source gives no resolution of the conjecture.