Gheri's conjecture on the p-Frobenius ratio

Let $G$ be a finite group and let $p$ be a prime dividing $|G|$. Define

$$\mathfrak{U}_p(G)=\bigcup_{P\in \operatorname{Syl}_p(G)}P$$

to be the set of $p$-elements of $G$. If $P$ is a Sylow $p$-subgroup of $G$, the positive integer $|\mathfrak{U}_p(G)|/|P|$ is called the $p$-Frobenius ratio of $G$, and let $n_p(G)$ denote the number of Sylow $p$-subgroups of $G$.

Gheri's conjecture. The $p$-Frobenius ratio satisfies

$$\frac{|\mathfrak{U}_p(G)|}{|P|}\geq n_p(G)^{1-\frac{1}{p}}.$$

The conjecture seeks a lower bound for the Frobenius ratio in terms of the number of Sylow $p$-subgroups. The ratio is known to be a positive integer by Frobenius' theorem, but its broader combinatorial meaning and the proposed bound remain open in general.