Cohn's SDPP conjecture for abelian groups

Let $G$ be a finite abelian group, and let $(A_i,B_i)_{i=1}^n$ be pairs of subsets satisfying the simultaneous double product property (SDPP): for each $i$, the equation $aa'^{-1}=bb'^{-1}$ has only the solution $a=a'$ and $b=b'$ for $a,a'\in A_i$ and $b,b'\in B_i$, and $a_i(a_j')^{-1}b_j(b_k')^{-1}=1$ implies $i=k$. Cohn's SDPP conjecture. For arbitrarily large $n$, there exists an abelian group $G$ with

$$|G|=n^{2-o(1)}$$

and $n$ pairs of sets $A_i,B_i$ satisfying the SDPP such that

$$|A_i||B_i|>n^{2-o(1)}.$$

The conjecture is motivated by the goal of obtaining matrix-multiplication exponent $\omega=2$ through SDPP constructions in abelian groups. It is presented here as an external conjecture, with no resolution supplied in the source.