The Bias Conjecture for second moments in one-parameter elliptic-curve families

Let $\mathcal{E}:y^2=x^3+A(T)x+B(T)$ be a family of elliptic curves over $\mathbb{Q}(T)$, with $A(T),B(T)\in\mathbb{Z}[T]$. For the specialization $E_t$, let $a_t(p)$ be its Fourier coefficient at $p$, and define

$$A_2(p)=\frac{1}{p}\sum_{t\bmod p}a_t(p)^2.$$

Write the second moment in the form

$$A_2(p)=p^2+\beta_{3/2}(p)p^{3/2}+\beta_1(p)p+\beta_{1/2}(p)p^{1/2}+\beta_0(p),$$

where each $\beta_r(p)$ is of order $1$.

Bias Conjecture. For any such family, the largest lower-order term in the second moment that does not average to $0$ is on average negative; equivalently, the first $\beta_r(p)$ term that does not average to zero has negative average.

Michel proved the leading estimate $A_2(p)=p^2+O(p^{3/2})$ for families with nonconstant $j$-invariant, and the possible lower-order sizes are $p^{3/2}$, $p$, $p^{1/2}$, and $1$. The conjecture is supported by the families proved or numerically analyzed in the source and is intended to describe a family-dependent arithmetic bias.