 Solution

Choose ${\aleph }_2$ distinct cofinal branches of the Kurepa tree and regard them as subsets of its underlying set ${\omega }_1$. Let $\mathcal{F}$ be this family. Fix an ordinal $\alpha <{\omega }_1$. The countably many node codes below $\alpha$ have heights bounded by some $\delta <{\omega }_1$. A cofinal branch meets level $\delta$, and its node there determines all its predecessors, hence all its nodes with codes below $\alpha$. Since level $\delta$ is countable,This is the Kurepa-family hypothesis. The given compatibility between the ordinal codes and the tree order ensures in particular that the branches are consistently viewed as subsets of ${\omega }_1$; boundedness of the heights of the countably many codes is what the argument uses.