Finiteness conjecture for exceptional heavenly elliptic curves over quadratic fields

For every real number $B\geq 1$, let $\mathcal{R}_{B}$ be the set of ordered pairs $(\ell,K)$ such that $K$ is a quadratic field, $\ell\geq B$ is prime, and there exists an elliptic curve $E/K$ heavenly at $\ell$ with

$$E\times_K\overline{\mathbb{Q}}\not\cong A\times_{\mathbb{Q}}\overline{\mathbb{Q}}$$

for every elliptic curve $A/\mathbb{Q}$ that is heavenly at $\ell$ over $\mathbb{Q}$. Finiteness conjecture. The set $\mathcal{R}_{7}$ is finite. This is the corresponding finiteness statement from the perspective of which quadratic fields admit heavenly elliptic curves not arising geometrically from heavenly elliptic curves over $\mathbb{Q}$; the paper proves its equivalence with the preceding finiteness conjecture.