The solvable-group supremum conjecture for odd exponents

Let $n$ be a positive integer, let $\mathcal{S}$ be the class of finite solvable groups, and for $H\in\mathcal{S}$ and $\phi\in\operatorname{Aut}(H)$ define

$$X_{n,\phi}(H):=\left\{x\in H\mid xx^\phi x^{\phi^2}\cdots x^{\phi^{n-1}}=1\right\}.$$

Solvable-group supremum conjecture. For every positive integer $n$, one has

$$c^{\mathcal S}_{2n+1}:=\sup\left(\left\{\frac{|X_{n,\phi}(H)|}{|H|}: H\in\mathcal S,\ \phi\in\operatorname{Aut}(H),\ \phi^{2n+1}=\operatorname{id}\right\}\setminus\{1\}\right)<1.$$

For odd exponents, the paper states that this conjecture is equivalent to the Lé vai–Pyber conjecture; no general proof is supplied.