I recently defined the almost-disjoint gap to be
$$\mathcal{D}_{\mu} \left( E_1, \ldots, E_n \right) \triangleq \mu \left( \bigcup_{k=1}^{n} E_k \right) - \sum_{k=1}^n \mu \left( E_k \right)$$
where $\mu$ is a measure and $\{ E_k \}_{k=1}^n$ are elements of a suitable $\sigma$-algebra.
Suppose that $\mu$ is finite so that there exists a lower bound for $\mathcal{D}_{\mu} \left( E_1, \ldots, E_n \right)$. In terms of the IE principle we know that $\mathcal{D}_{\mu} \left( E_1, \ldots, E_n \right)$ quantifies the joint terms among different combinations of events:
$$\mathcal{D}_{\mu} \left( E_1, \ldots, E_n \right) = \underbrace{\sum_{k=2}^n \left((-1)^{k-1} \sum_{\substack{I \subseteq \{1, \ldots, n \} \\ |I| = k}} \mu\left(\bigcap_{i \in I} E_i \right) \right)}_{\text{Joint Terms}}$$
Does $\mathcal{D}_{\mu} \left( E_1, \ldots, E_n \right)$ with finite $\mu$ achieve its lower bound if-and-only-if $E_1 = E_2 = \cdots = E_{n-1} = E_n$ almost surely?