Certain maximal objects whose existence follows from Zorn's Lemma have received some set-theoretic attention. Examples are maximal independent families and maximal almost disjoint families.

There is quite a bit of information out there about the two cardinal invariants $\mathfrak i$ and $\mathfrak a$, the minimal sizes of maximal independent families of subsets of $\omega$ and of maximal almost disjoint families of subsets of $\omega$.

However, I have not seen anything concerning maximal incomparable families: Let $\mathcal A$ be a family of subsets of $\omega$ that are both infinite and coinfinite. We call $\mathcal A$ incomparable if the elements of $\mathcal A$ are pairwise incomparable with respect to the relation $\subseteq^*$ where $A\subseteq^*B$ if $A\setminus B$ is finite. Note that both independent families and almost disjoint families are incomparable.

Question: What is the minimal size of a maximal incomparable family of subsets of $\omega$?

I am guessing that this cardinal invariant is equal to $2^{\aleph_0}$ or equal to $\mathfrak i$, but mostly for psychological/social reasons: It should have been looked at before, unless it is equal to one of the known invariants.