High rank quadratic twists of pairs of elliptic curves

Given a pair of elliptic curves E1 and E2 over the rational field Q whose j-invariants are not simultaneously 0 or 1728, Kuwata and Wang proved the existence of infinitely many square-free rationals d such that the d-quadratic twists of E1 and E2 are both of positive rank. We construct infinite families of pairs of elliptic curves E1 and E2 over Q such that for each pair there exist infinitely many square-free rationals d for which the d-quadratic twists of E1 and E2 are both of rank at least 2.