The abundance conjecture for torsion in Shafarevich–Tate groups

Let $k_0$ be the prime global subfield, and let $n,k\in\mathbb{Z}^+$ with $n>1$. For an elliptic curve $E$ over $k_0$, let ${\text{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}(k_0,E)$ denote its Shafarevich–Tate group.

Abundance conjecture for Shafarevich–Tate torsion. There exists an elliptic curve $E_{/k_0}$ such that ${\text{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}(k_0,E)$ has at least $k$ elements of order $n$.

The paper states that this conjecture implies the genus-one Hasse principle violation conjecture. For $k_0=\mathbb{Q}$, it reports known cases $n=2,3,5$, announced proofs for $n=7$ and $n=13$, and says that the remaining cases require new ideas; the general conjecture is therefore open.