Vanishing of low-degree Hilbert–Kunz coefficients under the standard conjecture

Let $R$ be a $d$-dimensional Cohen–Macaulay local ring of characteristic $p$ with perfect residue class field. Let $I$ be a maximal primary ideal of $R$ of finite projective dimension. For $n>0$, write

$$\ell_R(R/I^{[p^n]})=\sum_{i=0}^d\beta_i p^{in}.$$

Vanishing conjecture. If $i\le d/2$, then $\beta_i=0$.

This is motivated by the expected vanishing of the relevant numerical Chow groups in low degrees under Grothendieck's standard conjecture. No example is known in which the corresponding groups fail to vanish for some $i\le d/2$, and the stated coefficient-vanishing claim is presented as a natural question.