Invariance of log Lefschetz classes under prime-to-order powers

Let $f:V\to U$ be a finite étale morphism of connected separated and smooth schemes of finite type purely of dimension $d$ over a perfect field $F$, and let $\sigma$ be a non-trivial automorphism of $V$ over $U$. For an automorphism $\tau$ of $V$ over $U$, write $(\Gamma_\tau,\Delta_{\overline V})^{\log}$ for its log Lefschetz class, where $\Gamma_\tau\subset V\times_U V$ is the graph of $\tau$. Invariance conjecture. If $j$ is an integer prime to the order of $\sigma$, then

$$(\Gamma_\sigma,\Delta_{\overline V})^{\log}=(\Gamma_{\sigma^j},\Delta_{\overline V})^{\log}.$$

This is an expected symmetry of log Lefschetz classes under generators of the same cyclic subgroup; the supplied text gives no resolution, so the conjecture remains open.