The exact-order conjecture for hypercube saturation

For a positive integer $k$, let $Q_k$ be the $k$-dimensional hypercube viewed as a poset, and let $\operatorname{sat}^*(n,Q_k)$ denote the minimum size of a family of subsets of $[n]$ that is induced-$Q_k$-free but becomes non-free after adding any missing subset.

Hypercube saturation conjecture. For every $k\geq 2$,

$$\operatorname{sat}^*(n,Q_k)=(2^{k-1}-1)n-c,$$

for some absolute constant $c$.

The paper gives the construction $\operatorname{sat}^*(n,Q_3)\leq 3n-2$ and asks whether it is tight, motivating the proposed formula for all hypercubes. The general equality remains open.