The maximal-index conjecture for halidon rings of integers modulo n

Let $R=\mathbb{Z}_{n}$, where

$$n=p_{1}^{e_{1}}p_{2}^{e_{2}}p_{3}^{e_{3}}\cdots p_{k}^{e_{k}}$$

with primes $p_{1}<p_{2}<p_{3}<\cdots<p_{k}$, including $2$. Write $\psi(n)$ for the quantity appearing in the conjecture. Maximal-index conjecture. $R$ is a halidon ring with maximal index

$$m_{\max}=\psi(n).$$

The claim is presented as an empirical conclusion from running the author's computer programme on several values of $n$. Its status is not established in the supplied text.