On the conjecture of groups with the same number of centralizers

For any group Formula: see text, let Formula: see text denote the set of all centralizers of Formula: see text. K. Khoramshahi and M. Zarrin Groups with the same number of centralizers, J. Algebra Appl. 20 (2021) 2150012, posed the following conjecture: Let Formula: see text and Formula: see text be finite groups. Is it true that if Formula: see text and Formula: see text, then Formula: see text is isoclonic to Formula: see text? In this paper, among other things, we give a negative answer to this conjecture.