The determinant-character conjecture for finite abelian groups

Let $G$ be a finite abelian group, and let $A=\{a_1,\ldots,a_k\}$ and $B=\{b_1,\ldots,b_k\}$ be two $k$-subsets of $G$. Let $K$ be any field containing an element of multiplicative order $|G|$, and let $\widehat{G}$ be the character group of all group homomorphisms from $G$ to $K^*=K\setminus\{0\}$. The determinant-character conjecture. There are $\chi_1,\ldots,\chi_k\in\widehat{G}$ such that

$$\det(\chi_i(a_j))_{1\le i,j\le k}\ne0$$

and

$$\det(\chi_i(b_j))_{1\le i,j\le k}\ne0.$$

The conjecture is proposed as an exterior-algebra and character formulation whose validity would imply Snevily's conjecture for abelian groups of odd order; the source provides no resolution of it.