Coordinate-star extremal conjecture for intersecting families under product measures

Let $[n]=\{1,\ldots,n\}$, let $\Omega=2^{[n]}$, and let $\mu_{\bm p}$ be the product measure with coordinate probabilities $p^{(\ell)}$. A family $U\subset\Omega$ is intersecting if every two members of $U$ intersect.

Coordinate-star extremal conjecture. Assume

$$p^{(1)}=\max\{p^{(\ell)}:\ell\in[n]\},\qquad p^{(\ell)}\leqslant\frac12\quad\text{for }\ell\geqslant3.$$

Then every intersecting family $U\subset\Omega$ satisfies

$$\mu_{\bm p}(U)\leqslant p^{(1)}.$$

Moreover, if $p^{(1)}>p^{(\ell)}$ for $\ell\geqslant3$, or if $p^{(1)}<1/2$, equality holds if and only if

$$U=\{x\in\Omega:\ell\in x\}$$

for some $\ell\in[n]$ with $p^{(\ell)}=p^{(1)}$.

The conjecture seeks to extend the known product-measure bound by weakening the assumption from coordinates beginning at $\ell\geqslant2$ to those beginning at $\ell\geqslant3$. The stated equality cases identify the maximizing families as coordinate stars under the additional hypotheses.