Mennicke's modularity conjecture for elliptic curves over imaginary quadratic fields

Let $K$ be an imaginary quadratic field of class number $1$, and let $E$ be an elliptic curve over $K$ without complex multiplication by an order in $K$. Say that $E$ is modular if $L(E,s)=L(F,s)$ for some $F\in S_2(\operatorname{id}{n})$ over $K$, where $\operatorname{id}{n}$ is the conductor of $E$.

Mennicke's modularity conjecture. Every such elliptic curve $E$ is modular.

Some cases are known, and under mild hypotheses every such curve is potentially modular. Curves with complex multiplication are excluded because they correspond to Hecke characters and non-cuspidal modular forms.