Equal centralizer and commutator counts imply isoclinism

Let $G$ and $S$ be finite groups. Write $G'$ and $S'$ for their commutator subgroups, and let $cent(G)$ denote the set of centralizers of elements of $G$.

Centralizer–commutator conjecture. If

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

then $G$ is isoclinic to $S$.

The question is posed after counterexamples showing that equality of the numbers of centralizers alone does not imply isoclinism. The conjecture asks whether adding equality of the orders of the commutator subgroups is sufficient; no resolution is supplied in the source.