Totally balanced curves and complex multiplication

Determine whether, for a fixed prime \(\ell\) and a Galois extension \(K/\mathbb{Q}\) of odd degree, every elliptic curve over \(K\) that is totally balanced at \(\ell\) has complex multiplication.

Background

The paper shows that totally balanced elliptic curves over Galois extensions of $\mathbb{Q}$ of odd degree have trace behavior closely analogous to that of CM elliptic curves: their trace modulo $\ell$ lies in a proper subset of $\mathbb{F}_\ell$. It remains unresolved whether this similarity reflects an actual structural characterization, namely whether total balancedness forces complex multiplication.

The question is specifically posed for a fixed prime $\ell$ and a Galois number field of odd degree, and the paper emphasizes that this issue differs from the quadratic case because balancedness need not determine a unique representation type in higher degree.