The persistent genus-one Hasse principle violation conjecture over global field extensions

Let $k$ be a global field and let $k_0$ be its prime global subfield. A genus one curve $C$ violates the Hasse Principle over a field if it has points over every completion of that field but no rational point over that field.

Persistent genus-one Hasse principle violation conjecture. For every global field $k$, there is a genus one curve $C$ defined over $k_0$ such that $C$ violates the Hasse Principle over $k_0$ and also over $k$.

This asks for Hasse principle violations over the prime global subfield that remain violations after base extension to any global field. The supplied text presents it as a natural construction problem and gives no resolution, so it remains open.