Does there exist infinitely many quadratic field such that rank of elliptic curves gains at most 2?

Question

Let $L/K$ be a quadratic extension of number field. Most elliptic curves $E/K$ will have the property that $rank(E/K) \le rank(E/L) +2$.

Is it known for an arbitrary $E/K$, does there exists infinitely many $L/K$: quadratic such that $rank(E/L) \le rank(E/K) +2$?