A minimal Kurepa line

We show it is consistent with $\text{ZFC}$ that there is an everywhere Kurepa line which is order isomorphic to all of its dense ${\mathrm{\aleph }}_2$-dense suborders. Moreover, this Kurepa line does not contain any Aronszajn suborder. We also show it is consistent with $\text{ZFC}$ that there is a minimal Kurepa line which does not contain any Aronszajn suborder.