 Gap-doubling closed family of infinite subsets

The family $D=\{\{a_1<a_2<\cdots \}:a_{n+1}>2a_n\text{ for all }n\}$ is ordinarily closed and nowhere dense. A failed gap is witnessed by a finite prefix, and any cylinder can be refined by adjoining consecutive large integers that fail the gap. Nevertheless $D$ intersects every infinite-subset cone in continuum many sets: a binary tree chooses two sufficiently large next points inside its reservoir at each step.