Diophantine stability for elliptic curves on average

Let $K$ be a number field and $\ell \geq 5$ a prime number. Mazur and Rubin introduced the notion of diophantine stability for a variety $X_{/K}$ at a prime $\ell$. We show that there is a positive density set of elliptic curves $E_{/\mathbb{Q}}$ of rank $1$ such that $E_{/K}$ is diophantine stable at $\ell$. This has implications for Hilbert's Tenth Problem over $\mathscr{O}_K$. This problem asks whether there exists an algorithm that decides in finite time whether a finite system of Diophantine equations over $\mathscr{O}_K$ has a solution.