The finiteness conjecture for heavenly elliptic curves over quadratic fields

Let $\overline{\mathcal{H}}_{\geq 7}(2,1)$ denote the set defined in the paper, consisting of the relevant heavenly elliptic-curve data over quadratic fields with prime parameter at least $7$. Finiteness conjecture. The set

$$\overline{\mathcal{H}}_{\geq 7}(2,1)$$

is finite. The paper explains that this conjecture is implied by the balanced heavenly elliptic curve conjecture together with the uniform bound for heavenly elliptic curves with complex multiplication. Its status is not resolved in the supplied text.