Simultaneous double product property conjecture for abelian groups

Let $n$ be a positive integer. A collection of pairs of subsets $A_i,B_i$ of an abelian group $H$ satisfies the simultaneous double product property when the pairs satisfy the double product property and the corresponding difference-set disjointness conditions described in the paper. The parameters are measured by $|H|$ and the products $|A_i||B_i|$.

Simultaneous double product property conjecture. For arbitrarily large $n$, there exists an abelian group $H$ with $n$ pairs of subsets $A_i,B_i$ satisfying the simultaneous double product property such that

$$|H|=n^{2+o(1)}$$

and

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

If true, this would provide the algebraic route to the optimal matrix multiplication exponent $\omega=2$. The preceding parameter bounds show that $\alpha=\beta=2$ is the only possible way to achieve this exponent; existence of such families remains open.