Infinitely many rational elliptic curves with 2-torsion over quadratic fields

Let $p$ be a prime satisfying

$$p\equiv 11,19\pmod{20}.$$

For an elliptic curve $E/\mathbb Q$, write $E(\mathbb Q(\sqrt{-p}))_{\mathrm{tors}}$ for its torsion subgroup over $\mathbb Q(\sqrt{-p})$. Existence conjecture. There exist infinitely many elliptic curves $E/\mathbb Q$ such that

$$E(\mathbb Q(\sqrt{-p}))_{\mathrm{tors}}\cong \mathbb Z/2\mathbb Z\times\mathbb Z/10\mathbb Z.$$

This is one of the open questions left by the paper concerning prescribed torsion subgroups of rational elliptic curves over quadratic fields; the paper proves related non-existence results and establishes the asserted infinitude conditionally on finiteness of the $2$-primary part of the Tate–Shafarevich group.