Lower-bound conjecture for diamond-saturated families avoiding the extremes

Let $n$ be a positive integer, let $\mathcal{B}_n$ be the Boolean lattice, and let $\mathcal{F}\subseteq\mathcal{B}_n$ be an induced-$\mathcal{D}_2$-saturated family that avoids both $\emptyset$ and $[n]$. Lower-bound conjecture for diamond-saturated families avoiding the extremes. There exists a universal constant $c>0$ such that

$$2n-c\leq |\mathcal{F}|.$$

The conjecture seeks a stronger lower bound for induced-$\mathcal{D}_2$-saturated families avoiding the two extreme sets; the source gives a construction of size $2n$ and presents this lower bound as an open question.