Delaunay–Poonen–Rains conjecture for the density of elliptic curves with nontrivial Tate–Shafarevich group

Let $p$ be an odd prime. Let $\mathfrak{T}_p$ denote the set of elliptic curves $E_{/\mathbb{Q}}$ such that the $p$-primary part of the Tate–Shafarevich group $Sh(E/\mathbb{Q})[p^\infty]$ is nonzero, and let $\underline{\mathfrak{d}}$ and $\mathfrak{d}$ denote the lower and ordinary densities used above. A set $\mathcal{E}$ of elliptic curves is defined by local congruence conditions when membership is determined by finitely many local congruence conditions. Delaunay–Poonen–Rains conjecture. With respect to the notation above,

$$\underline{\mathfrak{d}}(\mathfrak{T}_p)\geq 1-\prod_{i\geq 1}\left(1-p^{-(2i-1)}\right)>p^{-1}+p^{-3}-p^{-4}.$$

Moreover, for every set $\mathcal{E}$ of elliptic curves defined by local congruence conditions,

$$\underline{\mathfrak{d}}(\mathfrak{T}_p\cap\mathcal{E})=\underline{\mathfrak{d}}(\mathfrak{T}_p)\mathfrak{d}(\mathcal{E}).$$

The claim records the predictions attributed to Delaunay and Poonen–Rains, based on Cohen–Lenstra heuristics and random matrix theory; the source assumes finiteness of the $p$-primary Tate–Shafarevich group, and no resolution is supplied here.