Equality of generalized commuting probabilities determines central quotient order

Let $G_{1}$ and $G_{2}$ be finite groups, and let $Pr^{2n}(G)$ denote the generalized commuting probability of $G$ at level $2n$. Write $Z(G)$ for the center of $G$. The conjecture. If

$$Pr^{2n}(G_{1})=Pr^{2n}(G_{2})$$

for every $n\geq0$, then

$$\left\vert G_{1}/Z(G_{1})\right\vert=\left\vert G_{2}/Z(G_{2})\right\vert.$$

The claim is motivated by the known fact that equality of all generalized commuting probabilities forces the commutator subgroups to have the same order. It is refuted by the groups $G_{2}$ and $G_{3}=D_{8}\times D_{8}$, which provide a counterexample to the opposite direction of the related isoclinism conjecture.