The centralizer-count and derived-order conjecture for finite groups

Let $G$ and $S$ be finite groups. Write $G'$ and $S'$ for their derived subgroups, and let $Cent(G)$ and $Cent(S)$ denote the sets of centralizers of elements of $G$ and $S$, respectively.

Centralizer-count and derived-order conjecture. If

$$|Cent(G)|=|Cent(S)|\quad\text{and}\quad |G'|=|S'|,$$

then $G$ is isoclinic to $S$.

The conjecture was posed as a question by the authors of the cited source. It is refuted: Zarrin gave a counterexample, and also showed that isoclinic finite groups have the same number of centralizers, whereas the converse fails.