Is the product of centers of centralizers a subgroup in general?

Determine whether, for an arbitrary finite group G and elements a1,...,an in G \ Z(G), the set product ∏_{i=1}^n Z_G(a_i), where Z_G(a)=Z(C_G(a)), is necessarily a subgroup of G without additional hypotheses (such as the ai commuting).

Background

In Section 3, the paper studies subgroups generated by the centers of centralizers of elements, defining Z_G(a)=Z(C_G(a)). These objects underpin characterizations of maximal abelian subgroups and several lemmas rely on manipulating products of such centers.

Before proving Lemma 3.5, the authors note that it is not generally established whether the product ∏ Z_G(a_i) forms a subgroup for arbitrary choices of noncentral elements. Because this point is unresolved in general, they add the subgroup property as an explicit assumption in Lemmas 3.5 and 3.6, while observing it holds when the elements a_i commute.