Tokushige's product-measure conjecture for cross $t$-intersecting families

Let $n\geq t\geq 1$ be integers. For $p\in(0,1)$, let $\mu_p$ be the product measure on $2^{[n]}$, defined by

$$\mu_p(F)=p^{|F|}(1-p)^{n-|F|}$$

for $F\subseteq[n]$, and extend it to families by $\mu_p(\mathcal F)=\sum_{F\in\mathcal F}\mu_p(F)$. Two families $\mathcal F_1,\mathcal F_2\subseteq2^{[n]}$ are cross $t$-intersecting if $|F_1\cap F_2|\geq t$ for every $F_1\in\mathcal F_1$ and $F_2\in\mathcal F_2$.

Tokushige's conjecture. Let $p_1,p_2\in(0,\frac{1}{t+1})$. If $\mathcal F_1$ and $\mathcal F_2$ are cross $t$-intersecting, then

$$\mu_{p_1}(\mathcal F_1)\mu_{p_2}(\mathcal F_2)\leq(p_1p_2)^t,$$

with equality if and only if

$$\mathcal F_1=\mathcal F_2=\{F\subseteq[n]:T\subseteq F\}$$

for some $T\in\binom{[n]}{t}$.

This conjecture extends the measure version of the Complete Intersection Theorem and is still widely open in the stated two-parameter form. The bound is known for $t=1$, several ranges with $p_1=p_2$, and, according to the source, the paper proves the inequality for all $p_1,p_2\in(0,\frac{1}{t+1})$ when $t\geq3$; the equality characterization is part of the conjecture.