Tolue's prime-degree non-centralizer graph conjecture

Let $G$ be a finite group, let $Z(G)$ be its center, and let $\Upsilon_G$ be the simple graph whose vertices are the elements of $G$, with distinct vertices $x$ and $y$ adjacent when $C_G(x)\neq C_G(y)$, where $C_G(x)$ is the centralizer of $x$ in $G$. A graph is $p$-regular if every vertex has degree $p$, where $p$ is prime. Tolue's conjecture. The graph $\Upsilon_G$ is not $p$-regular for any prime integer $p$. The conjecture concerns restrictions on the possible regular degrees of non-centralizer graphs; the source attributes it to Tolue and records the cases of degrees $4,5,6,7,8,11,13$, with degree $6$ occurring precisely for $D_8$ and $Q_8$.