Shumyatsky's conjecture on finite-rank profinite groups with bounded centralizers

Let $G$ be a profinite group. A profinite group has finite rank if there is an integer $r$ such that every finitely generated closed subgroup can be generated by at most $r$ elements. For each nontrivial element $g\in G$, write $C_G(g)$ for its centralizer.

Shumyatsky's conjecture. If $C_G(g)$ has finite rank for every nontrivial $g\in G$, and $G$ is not a pro-$p$ group, then $G$ has finite rank.

This conjecture concerns the extent to which restrictions on centralizers control the global generation rank of a profinite group. The paper's abstract states a stronger bounded-rank result, but the supplied text does not explicitly state whether this conjecture is thereby resolved.