A positive density of elliptic curves are diophantine stable in certain Galois extensions

Let $p\in \{3,5\}$ and consider a cyclic $p$-extension $L\mathrm{/}\mathbb{Q}$. We show that there exists an effective positive density of elliptic curves $E$ defined over $\mathbb{Q}$, ordered by height, that are diophantine stable in $L$.