Alternating-group conjugacy-class intersection conjecture

Let $G=A_n$ with $n$ odd, let $C$ be a conjugacy class of $n$-cycles, and let $H$ be the centralizer in $G$ of an element of $C$. Let $N_G(H)$ denote the normalizer of $H$ in $G$, and let $\phi$ be Euler's totient function. Alternating-group intersection conjecture. Then

$$\max\{|gH\cap C|\mid g\in G\}=|H\cap C|=|N_G(H):H|\in\{\phi(n),\phi(n)/2\}.$$

The claim would give a sharp bound for the quantity governing intersections of conjugacy classes with cosets of centralizers, in the alternating-group case. The source presents it as a conjecture and gives no resolution status.