Nonvanishing of 2-primary Kolyvagin classes via simultaneous quadratic twists

Let $E/\mathbf{Q}$ be an elliptic curve, let $K=\mathbf{Q}(\sqrt{D})$ satisfy the Heegner hypothesis with $D<-4$, and fix a modular parametrization $φ$ with $φ(\infty)=O$. Under surjectivity modulo $2^L$, with $L\ge 2$, and a cohomological exclusion condition at level $4$, we construct a finite set