Prescribed ranks over number fields beyond rank 1

Determine, for each fixed integer n ≥ 2 and any number field K, whether there exists an elliptic curve E/K whose Mordell–Weil group has rank exactly n.

Background

The authors discuss the general scarcity of results ensuring that specific ranks occur over arbitrary number fields. While they prove that rank 1 occurs infinitely often over any number field (settling a folklore conjecture), analogous existence statements for higher ranks are not known.

This question is distinct from distributional conjectures (such as PPPW or Goldfeld) and concerns the existence of curves of any prescribed rank over fixed number fields, thereby extending the scope of their main theorem from rank 1 to higher ranks.