The genus-one Hasse principle violation conjecture

Let $k$ be a global field. A genus one curve over $k$ is a smooth projective curve of genus one, and an elliptic curve $E$ over $k$ is a genus one curve with a specified $k$-rational point. Write ${\text{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}(k,E)$ for the Shafarevich–Tate group of $E$. A curve violates the Hasse Principle if it has points over every completion of $k$ but no $k$-rational point.

Genus-one Hasse principle violation conjecture. For every global field $k$, there exists a genus one curve $C_{/k}$ which violates the Hasse Principle. Equivalently, there exists an elliptic curve $E_{/k}$ with ${\text{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}(k,E) \neq 0$.

The conjecture asks whether every global field admits a locally soluble but globally insoluble genus one curve. The supplied text gives no resolution, so its status remains open.