Conjecture on the parity of Tate–Shafarevich groups of quadratic twists

Let $E$ be an elliptic curve as above, let $N$ be a prime with $N\equiv 3\pmod{4}$, and suppose $\beta_{-N}\ne 0$. Let $E^{-N}$ denote the quadratic twist of $E$ associated with the fundamental discriminant $-N$, and let \cyr X$(E^{-N})$ be its Tate–Shafarevich group. Tate–Shafarevich parity conjecture. If $a_E(2)$ is even, then

$$|\text{\cyr X}(E^{-N})|\text{ is even}.$$

The conjecture links the parity of a modular eigenvalue to the order of the Tate–Shafarevich group of a quadratic twist. It is motivated by the preceding coefficient-parity conjecture and remains open in the supplied text.