Rank-preserving elliptic curves conjecture for arbitrary number-field extensions

Let $K\subset L$ be number fields. Let $d\in\mathcal O_K$ be nonzero and satisfy the hypotheses of Theorem~--in particular, the hypotheses referred to as -- in the source. For the corresponding elliptic curve $E$, the rank-preserving elliptic curves conjecture. There exists such a $d$ for which

$$\operatorname{rk}E(L)=\operatorname{rk}E(K)\geq 1.$$

The assertion would produce positive-rank elliptic curves whose rank does not increase over $L$, thereby implying the relevant Diophantine-definability consequences; it is presented as an explicit conjectural criterion and is unresolved in the source.