**Van Den Berg-Kesten-Reimer inequality**

Let $n$ be a positive integer. For $i\in[n]$, let $\Omega_i$ be a finite set and $\mu_i$ a probability measure on it. Set $\Omega=\Omega_1\!\times\!\ldots\!\times\!\Omega_n$ and $\mu=\mu_1\!\times\!\ldots\!\times\!\mu_n$.
For $A\!\subset\!\Omega$ and $\sigma\!\subset\![n]$, let
$$
A_\sigma=\lbrace\omega\!\in\!A\!:\,\forall\upsilon\!\in\!\Omega,\,(\forall i\!\in\!\sigma,\,\upsilon_i\!=\!\omega_i)\!\implies\!\upsilon\!\in\!A\rbrace
$$
In other terms, *the occurrence of the event $A_\sigma$ is solely controlled by $\sigma$*.
Given two events $A,B\subset\Omega$, the notion of *disjoint occurrence of $A$ and $B$* is defined by the following event

$$
A\!\circ\!B=\lbrace\omega\!\in\!A\!\cap\!B\!:\,\exists\,\sigma,\tau\!\subset\![n],\,\sigma\!\cap\!\tau\!=\!\emptyset \wedge\omega\!\in\!A_\sigma\!\cap\!B_\tau\rbrace
$$

*Van Den Berg-Kesten-Reimer* inequality states that

$$ \forall A,B\!\subset\!\Omega,\,\,\mu(A\circ B)\le\mu(A)\cdot\mu(B) $$

**Question**

Are there non-trivial events that turn the inequality stated above into an equality?