Finite-rank conjecture for profinite groups with finite-rank centralizers

Let $G$ be a profinite group in which the centralizer of every nontrivial element has finite rank. Suppose that $G$ is not a pro-$p$ group.

Finite-rank conjecture. Then $G$ has finite rank.

This conjecture extends the preceding result for profinite groups whose nontrivial-element centralizers are virtually procyclic. It asks whether the same finite-rank conclusion remains valid under the weaker assumption that all such centralizers merely have finite rank.