Centralizer quotient conjecture for groups of finite asymptotic dimension

Let $G$ be a discrete group with finite asymptotic dimension, and for $g\in G$ let $C_g<G$ denote its centralizer. Centralizer quotient conjecture. The quotient

$$C_g/\langle g\rangle$$

also has finite asymptotic dimension. This is the second conjecture proposed, together with finite rational homological dimension, to imply the Burghelea conjecture in finite asymptotic dimension. The paper verifies it for many classes of groups, but the general assertion remains open.