Fifth-moment rank bias conjecture for elliptic-curve families

Consider a one-parameter family of elliptic curves with rank $r$. Let $A_{5,\mathcal{E}}(p)$ denote its fifth moment, whose main term has order $p^3$.

Fifth-moment rank bias conjecture. The average value of the main term of the fifth moment is $-5rp^3$.

The conjecture is part of a proposed relationship between average leading coefficients of odd moments and the rank of the family. The source bases it on numerical data and gives no proof; it remains open.