The equality characterization for the biased cross-intersection theorem

Let $\Omega$ be the parameter domain used in Theorem~, and let $\mu_\alpha$ and $\mu_\beta$ denote the corresponding biased measures on $2^{[n]}$. Let $\mathcal A_j$ and $\mathcal B_j$ be the families defined before that theorem. Assume that $\alpha$ and $\beta$ satisfy the premises of Theorem~, namely

$$\mu_\alpha(\mathcal A_j)\mu_\beta(\mathcal B_j)<\alpha\beta$$

for every $j\geq0$. The equality characterization. If $\mathcal F$ and $\mathcal G$ are cross-intersecting families in $2^{[n]}$ satisfying $\mu_\alpha(\mathcal F)\mu_\beta(\mathcal G)=\alpha\beta$, then there is some $i\in[n]$ such that

$$\mathcal F=\mathcal G=\{F\in2^{[n]}:i\in F\}.$$

This is a uniqueness assertion for equality in the biased cross-intersection bound established by the preceding theorem; because the supplied text gives no indication that this conjectural equality characterization has been proved, its resolution remains open.