The norm relations test over arbitrary number fields

Let $E / L$ be an elliptic curve, let $F / L$ be a Galois extension with Galois group $G$, and let $\rho$ be an irreducible representation of $G$. Suppose that

$$N_{\mathbb{Q}(\rho) / \mathbb{Q}}(\rho)^{\oplus m} = \left( \bigoplus_{\gamma \in \operatorname{Gal}(\mathbb{Q}(\rho) / \mathbb{Q})} \rho^\gamma \right)^{\oplus m} = \left( \bigoplus_i \mathbb{C}[G / H_i] \right) \ominus \left( \bigoplus_j \mathbb{C}[G / H_j'] \right)$$

for some $m \in \mathbb{Z}$ and subgroups $H_i,H_j' \leq G$. Norm relations test. If either

$$\frac{\prod_i C_{E / F^{H_i}}}{\prod_j C_{E / F^{H_j'}}}$$

is not a norm from some quadratic field $\mathbb{Q}(\sqrt{D}) \subset \mathbb{Q}(\rho)$, or is not a rational square when $m$ is even, then $E$ has a point of infinite order over $F$. This is the extension of the norm relations test from elliptic curves over $\mathbb{Q}$ to arbitrary number fields. The paper states that it follows from the parity conjecture for twists under the relevant semistable-reduction hypotheses; no unconditional resolution is given.