Serre's character conjecture for local automorphism groups

Let $A$ be a regular local noetherian ring and let $G$ be a finite group of automorphisms of $A$. Assume that the fixed part $A^G$ is noetherian and that, for every $\sigma\in G$ with $\sigma\neq 1$, the quotient $A/I_\sigma$ has finite length, where

$$I_\sigma=\langle \sigma(a)-a\mid a\in A\rangle.$$

Define the $\mathbb Z$-valued function $a_G$ on $G$ by

$$a_G(\sigma)= \begin{cases} \operatorname{length}(A/I_\sigma)&\text{if }\sigma\neq 1,\\ -\displaystyle\sum_{\tau\in G,\,\tau\neq 1}a_G(\tau)&\text{if }\sigma=1. \end{cases}$$

Serre's character conjecture. The function $a_G$ is a character of $G$. The supplied text gives no resolution status.