Existence conjecture for elliptic curves and imaginary quadratic fields at large primes

Let $p$ be a prime. For an elliptic curve $E/\mathbb{Q}$ and an imaginary quadratic field $K_0$, consider the triple $(E,K_0,p)$ and the conditions of the source's density lemma for constructing imaginary quadratic fields with the stated Hilbert-tenth property.

Large-prime existence conjecture. For all sufficiently large primes $p$, there exist an elliptic curve $E/\mathbb{Q}$ and an imaginary quadratic field $K_0$ such that the triple $(E,K_0,p)$ satisfies all the conditions of that density lemma.

The conjecture is motivated by explicit examples for $p=11,13,31,37$, which yield positive-density results. No resolution is given for all sufficiently large primes.