A Family of Elliptic Curves with a Lower Bound on 2-Selmer Ranks of Quadratic Twists

For any number field K with a complex place, we present an infinite family of elliptic curves defined over K such that $dim \mathbb{F}_2 Sel_2(E^F/K) \ge dim \mathbb{F}_2 E^F(K)[2] + r_2$ for every quadratic twist E^F of every curve E in this family, where r_2 is the number of complex places of K. This provides a counterexample to a conjecture appearing in work of Mazur and Rubin.