Eichler–Shimura-type conjecture for weight two eigenforms over number fields

Let $K$ be a number field, let $\mathfrak N$ be an ideal of $\mathcal O_K$, and let $\mathfrak f$ be a non-trivial, new, weight two complex eigenform over $K$ of level $\mathfrak N$. A fake elliptic curve is a simple abelian surface whose algebra of $K$-endomorphisms is an indefinite division quaternion algebra over $\mathbb Q$. Eichler–Shimura-type conjecture. If $K$ has a real place, there exists an elliptic curve $E_{\mathfrak f}/K$ of conductor $\mathfrak N$ such that

$$\#E_{\mathfrak f}(\mathcal O_K/\mathfrak q)=1+\operatorname{Norm}(\mathfrak q)-\mathfrak f(T_{\mathfrak q})$$

for every $\mathfrak q\nmid\mathfrak N$. If $K$ is totally complex, there exists either such an elliptic curve or a fake elliptic curve $A_{\mathfrak f}/K$ of conductor $\mathfrak N^2$ satisfying

$$\#A_{\mathfrak f}(\mathcal O_K/\mathfrak q)=(1+\operatorname{Norm}(\mathfrak q)-\mathfrak f(T_{\mathfrak q}))^2$$

for every $\mathfrak q\nmid\mathfrak N$. This is a special case of a fundamental conjecture from the Langlands programme concerning the connection between eigenforms and elliptic or fake elliptic curves; the source gives no resolution status.