 Kurepa-family hypothesis

There is $\mathcal{F}\subseteq \mathcal{P}({\omega }_1)$ of size ${\aleph }_2$ such that $\{X\cap \alpha :X\in \mathcal{F}\}$ is countable for every $\alpha <{\omega }_1$. Its initial-segment tree, pruned and leveled to be normal, is a Kurepa tree.