Farrokhi–Safa conjecture on relative commutativity degrees and central quotients

Let $G$ be a finite group and $H$ a subgroup of $G$. Write

$$\mathcal{D}(G)=\{d_0,d_1,\ldots,d_n\},\qquad 1=d_0>d_1>\cdots>d_n=d(G),$$

and let $d(H,G)$ denote the relative commutativity degree of $H$ in $G$. Define $Z(H,G)$ as the subgroup of elements of $H$ that commute with every element of $G$.

Farrokhi–Safa's conjecture. If $d(H,G)=d_k$, then $|H/Z(H,G)|$ is a product of at most $k$ primes. Consequently, $|G/Z(G)|$ is a product of at most $|\mathcal{D}(G)|-1$ primes.

This conjecture extends known classifications of finite groups whose sets of commutativity degrees have at most four elements. The cited results establish the analogous bound for certain initial relative commutativity degrees, but the general assertion is presented as a conjecture.