Let $[\omega]^\omega$ denote the collection of infinite subsets of $\omega$. We say $a,b\in [\omega]^\omega$ are *almost disjoint* if $a\cap b$ is finite. A subset $A\subseteq [\omega]^\omega$ is said to be an *almost disjoint (AD) family* if $a, b$ are almost disjoint for all $a \neq b \in A$. Zorn's Lemma shows that every almost disjoint family is contained in a *maximal almost disjoint (MAD) family* (maximal with respect to $\subseteq$).

If ${\cal A}$ is an AD family, we say that ${\cal A}$ is *of true cardinality ${\frak c}$* if for every $X\in [\omega]^\omega$ the set $\{A\in {\cal A}: |A\cap X| = \aleph_0\}$ is either finite or of size ${\frak c}$.

**Question.** Let ${\cal A}$ be an AD family, and let ${\cal M}$ be a MAD family such that ${\cal A}\subseteq {\cal M}$. If ${\cal A}$ is of true cardinality ${\frak c}$, is ${\cal M}$ necessarily of true cardinality ${\frak c}$?