Delaunay's moment conjecture for Tate–Shafarevich groups of elliptic curves

Let $\ell$ be a positive integer, let $\lambda=1^{m_1}2^{m_2}\cdots \ell^{m_\ell}$ be an integer partition, and let $u\geqslant0$. Consider elliptic curves $E/\mathbb Q$ of rank $u$, ordered by conductor, and write $\text{\tencyr\cyracc{Sh}}(E)[p^j]$ for the subgroup of the Tate–Shafarevich group killed by $p^j$.

Tate–Shafarevich moment conjecture. The average of

$$|\text{\tencyr\cyracc{Sh}}(E)[p]|^{m_1}|\text{\tencyr\cyracc{Sh}}(E)[p^2]|^{m_2}\cdots|\text{\tencyr\cyracc{Sh}}(E)[p^\ell]|^{m_\ell}$$

over these curves is

$$\sum_{\mu\subseteq\lambda}C_{\lambda,\mu}(p^2)p^{-|\mu|(2u-1)}.$$

This is the corrected Delaunay heuristic for Tate–Shafarevich groups, motivated by numerical evidence and compatibility with Selmer-group results; the asserted averages are generally open.