Suda–Tanaka–Tokushige's stability conjecture for cross-intersecting families

Let $[n]=\{1,\ldots,n\}$, let $\mathbf p=(p_1,\ldots,p_n)$ and $\mathbf q=(q_1,\ldots,q_n)$ lie in $(0,\frac12)^n$, and let $\mu_{\mathbf p}$ and $\mu_{\mathbf q}$ be the corresponding product measures on $2^{[n]}$. For $j\in[n]$, define the star

$$\mathcal S_j=\{A\subseteq[n]:j\in A\}.$$

Families $\mathcal A,\mathcal B\subseteq2^{[n]}$ are cross-intersecting if $A\cap B\neq\varnothing$ for every $A\in\mathcal A$ and $B\in\mathcal B$.

Suda–Tanaka–Tokushige's stability conjecture. Assume

$$p_1=\max_{j\in[n]}p_j\qquad\text{and}\qquad q_1=\max_{j\in[n]}q_j.$$

Then there is a constant $c=c(p_1,q_1)$ such that, whenever $\mathcal A$ and $\mathcal B$ are cross-intersecting and

$$\mu_{\mathbf p}(\mathcal A)\mu_{\mathbf q}(\mathcal B)>(1-\varepsilon)p_1q_1,$$

there exists $j\in[n]$ such that

$$\max\bigl\{\mu_{\mathbf p}(\mathcal A\mathbin{\triangle}\mathcal S_j), \mu_{\mathbf q}(\mathcal B\mathbin{\triangle}\mathcal S_j)\bigr\}<c\sqrt\varepsilon.$$

The result asserts quantitative closeness to a common star for near-extremal pairs. The source presents it as a conjecture following its theorem confirming the product inequality; no resolution is supplied here.