Bialostocki's zero-sum permutation conjecture

Let $n$ be a positive even integer, and let $a_1,\ldots,a_n$ and $w_1,\ldots,w_n$ be zero-sum sequences with terms in $\mathbb{Z}_n=\mathbb{Z}/n\mathbb{Z}$. Thus

$$\sum_{k=1}^n a_k=\sum_{k=1}^n w_k=0.$$

Bialostocki's conjecture. There exists a permutation $\sigma\in S_n$ such that

$$\sum_{k=1}^n w_k a_{\sigma(k)}=0,$$

where $S_n$ is the symmetric group on $\\{1,\ldots,n\\}$. The conjecture concerns pairing two zero-sum sequences so that their weighted inner product vanishes modulo $n$; the paper confirms it when the weights form an arithmetic progression with even common difference.