The minimalist conjecture for elliptic-curve Artin twists

Let $F/\mathbb{Q}$ be a Galois extension and let $\rho$ be an irreducible Artin representation factoring through $F/\mathbb{Q}$. Write $\langle\rho,E(F)\otimes_{\mathbb Z}\mathbb C\rangle$ for its multiplicity, and let $w(E/\mathbb Q,\rho)$ be the corresponding twisted global root number.

Minimalist conjecture for twists. For $100\%$ of elliptic curves $E/\mathbb Q$,

$$\langle \rho,E(F)\otimes_{\mathbb Z}\mathbb C\rangle= \begin{cases} 0 & \text{if }\rho\text{ is not self-dual},\\ 0 & \text{if }w(E/\mathbb Q,\rho)=+1,\\ 1 & \text{if }w(E/\mathbb Q,\rho)=-1. \end{cases}$$

The conjecture predicts the smallest possible multiplicity compatible with the parity conjecture for twists, for almost all elliptic curves in the stated ordering. It is presented as folklore and remains open.