The balanced heavenly elliptic curve conjecture

Let $\ell$ be a prime, let $K$ be a quadratic number field, and let $E/K$ be an elliptic curve. An elliptic curve is heavenly at $\ell$ when its $\ell$-power torsion field satisfies the heavenly condition defined in the paper, and it is balanced when its $\ell$-torsion has the balanced group-scheme structure defined there. Balanced heavenly elliptic curve conjecture. If $E/K$ is heavenly at $\ell$ and balanced, then $E$ has complex multiplication. The conjecture would imply finiteness of the relevant set of heavenly elliptic curves over quadratic fields, in view of the known uniform bound in the complex-multiplication case. Its status is not resolved in the supplied text.