Three-fold cross-intersection product-measure conjecture

Let $n$ be a positive integer and let $p\leq\frac23$. Set ${\bm p}=(p,p,\ldots,p)\in(0,1)^n$. For $i=1,2,3$, let $\mathcal A_i\subset 2^{[n]}$. The families are 3-cross intersecting if

$$A_1\cap A_2\cap A_3\neq\emptyset$$

for every $A_i\in\mathcal A_i$. Let $\mu_{\bm p}$ be the product biased measure on $2^{[n]}$.

Three-fold cross-intersection product-measure conjecture. If $\mathcal A_1,\mathcal A_2,\mathcal A_3$ are 3-cross intersecting, then

$$\mu_{\bm p}(\mathcal A_1)\mu_{\bm p}(\mathcal A_2)\mu_{\bm p}(\mathcal A_3)\leq p^3.$$

The text identifies this as an open problem and describes it as the easiest unresolved case for three cross-intersecting families; almost nothing is known for $r\geq3$.