We prove that it is consistent that there exists a Kurepa tree $T$ such that ${}^{ω_1}2$ is a continuous image of the topological space $[T]$ consisting of all cofinal branches of $T$ with respect to the cone topologies. This result solves an open problem due to Bergfalk, Chodounský, Guzmán, and Hrušák. We also prove that any Kurepa tree with the above property contains an Aronszajn subtree.