On the conjecture of groups with the same number of centralizer

For any group G, let $cent(G)$ denote the set of all centralizers of G. The authors in {KZ}, Groups with the same number of centralizers, J. Algebra Appl. (2021) 2150012 (6 pages), posed the following conjecture: Let G and S be finite groups. Is it true that if $|Cent(G)|=|Cent(S)|$ and $|G'|=|S'|$, then G is isoclonic to S? In this paper, among other things, we give a negative answer to this conjecture.