Catalan-number conjecture for odd moments of elliptic-curve families

Consider a one-parameter family of elliptic curves with rank $r$. Let $A_{2k+1,\mathcal{E}}(p)$ denote the $(2k+1)$st moment, and let $C_n$ be the $n$th Catalan number:

$$C_n=\frac{1}{n+1}\binom{2n}{n}.$$

For $k\in\mathbb{Z}_{>0}$, the proposed leading term has order $p^{k+1}$.

Catalan-number odd-moment conjecture. The average value of the main term of the $(2k+1)$st moment is

$$-C_{k+1}rp^{k+1}.$$

This conjecture unifies the proposed formulas for the third, fifth, and seventh moments through Catalan numbers. It is motivated by numerical data and is not proved in the source; it remains open.