Arithmetic properties of solubilizers

Let $G$ be a finite group and $x\in G$. In 2013, Hai-Reuven proved that the cardinality of the centralizer of $x$ divides the cardinality of the solubilizer of $x$. Settling a conjecture of Mousavi, Poozesh and Zamani, we show that the cardinality of the normalizer of $⟨x⟩$ is also a divisor. We actually give direct short proofs of more general results.