Generalized Lang–Trotter conjecture for coinciding Frobenius fields

Let $E_1$ and $E_2$ be elliptic curves over the rationals without complex multiplication. For $x>0$, define

$$S(x,E_1,E_2):=\#\{p\leq x\mid F(E_1,p)=F(E_2,p)\},$$

where $F(E_i,p)$ denotes the Frobenius field of $E_i$ at the prime $p$.

Generalized Lang–Trotter conjecture. The elliptic curves $E_1$ and $E_2$ satisfy

$$E_1\text{ is not isogenous to }E_2$$

if and only if

$$S(x,E_1,E_2)=O\left(\frac{\sqrt{x}}{\log x}\right).$$

This conjecture asks how often two non-CM elliptic curves have the same Frobenius field. It is presented as a generalized version of a question suggested by Lang and Trotter; the statement is not asserted as proved in the source and remains open.