Sun's permutation conjecture for nonvanishing products modulo n

Let $n>1$ be an integer, and let $D(n)$ denote the set of positive divisors of $n$. For integers $m_1,\ldots,m_{n-1}$, assume that

$$\left|\{1\leqslant s<n:\ d\nmid m_s\}\right|\geqslant d-1\qquad\text{for every }d\in D(n).$$

Sun's conjecture. There exists a permutation $\sigma$ of $\{1,\ldots,n-1\}$ such that $n\nmid s m_{\sigma(s)}$ for every $s=1,\ldots,n-1$. Theorem 1.1 in the paper confirms this conjecture in the cyclic-group case.