Dasgupta–Kárólyi–Serra–Szegedy conjecture for distinct products in finite abelian groups

Let $G$ be a finite abelian group with $|G|>1$, and let $p$ be the smallest prime divisor of $|G|$. Let $k<p$, let $a_1,\ldots,a_k\in G$ be pairwise distinct, and let $b_1,\ldots,b_k\in G$. Dasgupta–Kárólyi–Serra–Szegedy conjecture. There is a permutation $\sigma\in S_k$ such that

$$a_1b_{\sigma(1)},\ldots,a_kb_{\sigma(k)}$$

are pairwise distinct. This extends the corresponding theorem for cyclic groups of prime-power order and elementary abelian groups. The conjecture is presented as a proposed extension of those results; the supplied text gives no resolution status.