Is there an almost disjoint family $\mathcal{F}$ of subsets of $\omega$ of cardinality $2^{\aleph_0}$ satisfying the following property?

For all $A,B\in\mathcal{F}$ with $A\neq B$ and every $k\in\omega$, there is an $n\in\omega$ such that $|a-b|\geqslant k$ for all $a\in A\setminus n$ and $b\in B\setminus n$.