 Kurepa tree from an inaccessible binary tree

Let $\kappa$ be strongly inaccessible cardinal in the ground model and perform the finite Lévy collapse to omega-one. The ground full binary set-theoretic tree of height $\kappa$ has all levels of size less than $\kappa$, so these levels become countable. Its at least $({\kappa }^{+}{)}^M$ ground branches remain distinct. The chain condition for forcing preserves $({\kappa }^{+}{)}^M$, which is the extension ${\aleph }_2$. The unchanged ground set-theoretic tree therefore witnesses the Kurepa hypothesis.