Delaunay distribution conjecture for Shafarevich–Tate groups in fixed rank

Fix a global field $k$ and $r \in {\mathbb Z}_{\ge 0}$ such that ${\mathscr E}_r=\{E\in{\mathscr E}:\operatorname{rk}E(k)=r\}$ is infinite. For each finite abelian $p$-group $G$, consider the $p$-primary Shafarevich–Tate group $\Sha[p^\infty]$. Delaunay distribution conjecture. The density of

$$\{E \in {\mathscr E}_r: \Sha[p^\infty] \simeq G\}$$

in ${\mathscr E}_r$ equals the ${\mathscr T}_r$-probability of $G$. This is a fixed-rank refinement of the model and is identified with Delaunay's conjectural distribution for rank $r$ elliptic curves over $\mathbb Q$; the source gives no resolution.