Delaunay's moment conjecture for Tate–Shafarevich groups

Let $p$ be a prime. For a finite abelian $p$-group of type $\lambda$, let $C_{\lambda,\mu}(p^2)$ denote the corresponding subgroup-counting polynomial evaluated at $p^2$. Let ${\mathcal F}_u$ be the family of elliptic curves $E/\mathbb Q$ of Mordell–Weil rank $u$, ordered by conductor, and write $\text{\tencyr\cyracc{Sh}}(E)[p^j]$ for the subgroup of the Tate–Shafarevich group annihilated by $p^j$.

Delaunay's moment conjecture. For every positive integer $\ell$, partition $\lambda=1^{m_1}2^{m_2}\cdots \ell^{m_\ell}$, and nonnegative integer $u$, the average over $E\in{\mathcal F}_u$ satisfies

$$\operatorname{Avg}_{E\in{\mathcal F}_u}\,\prod_{j=1}^{\ell}|\text{\tencyr\cyracc{Sh}}(E)[p^j]|^{m_j}=\sum_{\mu\subseteq\lambda}C_{\lambda,\mu}(p^2)p^{-|\mu|(2u-1)}.$$

This extends Cohen–Lenstra-type moment predictions to Tate–Shafarevich groups in fixed-rank families of elliptic curves. The source states it as a conjecture and gives no resolution.