Stability conjecture for optimal cross-intersecting families

Let $\Omega_1=\Omega_2=2^{[n]}$, let $\mu_1,\mu_2$ be the product measures defined above, and let $U_i^{(\ell)}=\{x\in\Omega_i:\ell\in x\}$. Assume the condition on the coordinate probabilities in the original theorem holds, and suppose that $p_1,p_2<1/2$.

Stability conjecture. There exists a constant $c=c(p_1,p_2)$ such that, for every pair of cross-intersecting families $U_1\subset\Omega_1$ and $U_2\subset\Omega_2$ satisfying

$$\mu_1(U_1)\mu_2(U_2)>(1-\varepsilon)p_1p_2,$$

there is an $\ell\in[n]$ such that

$$\max\bigl\{\mu_1(U_1\triangle U_1^{(\ell)}),\mu_2(U_2\triangle U_2^{(\ell)})\bigr\}<c\sqrt{\varepsilon}.$$

The conjecture asserts quantitative closeness to an optimal coordinate-star pair when the product of measures is close to the extremal value. The source gives no resolution of this stability statement.