Rank-preserving elliptic curves conjecture for prime cyclic extensions

Let $K\subset L$ be a cyclic extension of prime degree of number fields, with $\sqrt{-1}\notin L$. Let $d\in\mathcal O_K$ be nonzero and satisfy the hypotheses of Theorem~--in particular, the hypotheses referred to as -- in the source. Consider

$$E:y^2=x^3+x+d^2.$$

The rank-preserving elliptic curves conjecture. There exists such a $d$ for which $(0,d)$ has infinite order and generates $E(L)\otimes\mathbb Q$. This would provide a rank-preserving positive-rank elliptic curve relevant to Diophantine definability and Hilbert's Tenth problem over rings of integers; the source gives no resolution.