The structural conjecture for minimum diamond-saturated families

Let $\mathcal D_2$ denote the diamond poset, and let $\mathcal F$ be a diamond-saturated family of subsets of $[n]$: it contains no induced copy of $\mathcal D_2$, while adding any missing subset creates one.

Structural conjecture for minimum diamond-saturated families. If

$$|\mathcal F|=n+1,$$

then $\mathcal F$ is either a maximal chain, the empty set together with all singletons, or the full set together with all complements of singletons. Moreover, if $\emptyset,[n] otin\mathcal F$, then

$$|\mathcal F|\geq 2n-c,$$

for some universal constant $c$.

The paper proves that every diamond-saturated family has size at least $n+1$, but the structure of families attaining equality is not determined. The asserted classification and the stronger lower bound without $\emptyset$ or $[n]$ remain open.