Profinite groups with restricted centralizers of powers

A group G is said to have restricted centralizers if for every \(x\in G\) the centralizer \(C_G(x)\) either is finite or has finite index in G. Shalev showed that a profinite group with restricted centralizers is virtually abelian. Here we take interest in profinite groups G for which there is an integer n such that \(C_G(x^n)\) is either finite or open whenever \(x\in G\). It is shown that such a group G has an open normal subgroup T with the property that G/Z(T) has finite exponent.