Bounded rank-preservation conjecture for elliptic curves over number-field extensions

Let $K\subset L$ be an extension of number fields. Let $\alpha\in\mathcal O_K$ and nonzero $d\in\mathcal O_K$ satisfy the hypotheses of Theorem~--in particular, the hypotheses referred to as -- in the source. For the corresponding elliptic curve $E$, the bounded rank-preservation conjecture. There exist $\alpha$ and $d$ such that

$$1\leq \operatorname{rk}E(K)=\operatorname{rk}E(L)\leq 2.$$

This generalizes the preceding rank-preservation formulation to arbitrary extensions while imposing an upper bound on the common rank; the source gives no proof or disproof.