Mordell-Weil Rank Growth of Elliptic Curves on Field Extensions

I am a graduate student checking to make sure his research is not already known.

Let $ mathbb {F} $ to be a numeric field and let $ E $ to be an elliptical curve defined on $ mathbb {F} $. Do we already know that for all $ r geq1 $there is an extension of finite degree $ mathbb {K} $ of $ mathbb {F} $ so that $ textrm {rank} left (E left ( mathbb {K} right) right) geq r $?