Squarefree-value conjecture for elliptic-curve sequences

Let $f(x,y)\in\mathbb{Z}[x,y]$ be squarefree, let $E$ be a non-CM elliptic curve defined over $\mathbb{Q}$, and let $\pi_{E,f}^{SF}(X)$ count the primes $p\leq X$ for which the associated value $f_p(E)$ is squarefree. Define $C_{E,f}^{SF}$ by

$$C_{E,f}^{SF}=\prod_{\ell\nmid M_E}\left(1-\frac{|C_f(\ell^2)|}{|\operatorname{GL}_2(\mathbb{Z}/\ell^2\mathbb{Z})|}\right)\sum_{n\mid M_E}\mu(n)\frac{|C_{E,f}(n^2)|}{|G_E(n^2)|}.$$

Squarefree-value conjecture. As $X\to\infty$,

$$\pi_{E,f}^{SF}(X)\sim C_{E,f}^{SF}\,\pi(X).$$

The preceding upper-bound theorem gives the correct order of magnitude and the conjectural constant, providing evidence for this asymptotic. The paper notes that the conjecture has been proved on average over the family of all elliptic curves for some specific sequences $f_p(E)$.