Density conjecture for Serre curves

Let $E/\mathbb{Q}$ be a Serre curve. Let $\Delta$ be the discriminant of $E$, let $D$ be the discriminant of $\mathbb{Q}(\sqrt{\Delta})$, and define

$$C=\prod_{\ell\text{ prime}}\left(1-\frac{\ell}{(\ell-1)^2(\ell+1)}\right).$$

Density conjecture. The density of primes $p$ for which $\operatorname{gcd}(\#E(\mathbb{F}_p),p-1)=1$ is

$$\begin{cases} C&\text{if }D\equiv0\pmod{4},\\ \left(1+\displaystyle\prod_{\ell\mid D}\frac{-\ell}{\ell^3-\ell^2-2\ell+1}\right)C&\text{if }D\equiv1\pmod{4}. \end{cases}$$

This is the expected answer to the density problem for Serre curves and is supported by the paper's main counting theorem; proving the required result when the set of auxiliary primes is allowed to contain all primes remains open.