Order of rank bias for rational elliptic curves

Let $p$ be a fixed prime. For each of the families $\mathcal E_{\mathrm{pr}}$, $\mathcal E_{\mathrm{all}}$, and $\mathcal E_{\mathrm{ht}}$, let $\mathcal E_r(X)$ denote the curves of rank $r$ with size less than $X$, and define

$$\mathcal A_r(p,X;\phi)=\frac{1}{\#\mathcal E_r(X)}\sum_{E\in\mathcal E_r(X),\,|E|<X,\,(N_E,p)=1}a_p(E)\phi(N_E).$$

Order of rank bias. If $r\geq0$ and $\mathcal E_r$ is infinite, then there exists $\delta>0$ such that, for every $\epsilon>0$, if $\phi(N)\ll(\log N)^{\delta-\epsilon}$ then $\mathcal A_r(p,X;\phi)\to0$, while if $\phi(N)\gg(\log N)^{\delta-\epsilon}$ then $|\mathcal A_r(p,X;\phi)|\to\infty$; moreover, the limit is $+\infty$ for $r\leq1$ and $-\infty$ for $r\geq2$. This asserts that the rank bias has inverse polylogarithmic order in the conductor, subject to the stated infinitude hypothesis.