Rigidity in equivariant algebraic $K$-theory

If $(R,I)$ is a henselian pair with an action of a finite group $G$ and $n\ge 1$ is an integer coprime to $|G|$ and such that $n\cdot |G|\in R^*$, then the reduction map of mod-$n$ equivariant $K$-theory spectra $$K^G(R)/n\stackrel{\simeq}{\longrightarrow} K^G(R/I)/n$$ is an equivalence. We prove this by revisiting the recent proof of non-equivariant rigidity by Clausen, Mathew, and Morrow.