For $A, B\subseteq \omega$ we write $A\subseteq^*B$ if $A\setminus B$ is finite. Let $[\omega]^\omega$ be the collection of infinite subsets.

We say that ${\cal D}\subseteq [\omega]^\omega$ is dominating from below if for all $X\in [\omega]^\omega$ there is $D\in {\cal D}$ with $D \subseteq^* X$. Let ${\frak D}$ be the collection of families in $[\omega]^\omega$ that are dominating from below.

Question. Is it consistent that $\min\{|{\cal D}|: {\cal D} \in {\frak D}\} < 2^{\aleph_0}$?