Chatterji–Mislin's centralizer formula for complete Euler characteristics

Let $G$ be a group of type FP over $\mathbb C$, and suppose that the centralizer of every finite-order element of $G$ has finite $L^2$-Betti numbers. Let $E(G)(s)$ denote the $s$-component of the complete Euler characteristic of $G$, let $C_G(s)$ be the centralizer of $s$ in $G$, and let $\chi^{(2)}(C_G(s))$ denote its $L^2$-Euler characteristic. The centralizer formula conjecture. For every $s\in G$,

$$E(G)(s)=\chi^{(2)}(C_G(s)).$$

The formula generalizes Brown's formula in the setting of $L^2$-homology and is stated as a proposed generalization for cases where Brown's formula is unavailable; the supplied text does not establish it in full generality.