Djanković–Ivan bounded-defect antichain saturation conjecture

For positive integers $k$ and $n$, let $\operatorname{sat*}(n,k)$ be the minimum size of a family $\mathcal{F}\subseteq2^{[n]}$ that contains no antichain of size $k$ and is maximal with respect to this property: adjoining any missing subset creates an antichain of size $k$.

Djanković–Ivan bounded-defect conjecture.

$$\operatorname{sat*}(n,k)=n(k-1)-O_k(1).$$

This strengthens the previously stated asymptotic conjecture by asserting that the deficit from $n(k-1)$ is bounded in terms of $k$ alone. The source says that the conjecture was proposed after exact results for $k=5$ and $k=6$, while its general validity remains open.