Parity conjecture for the square-index coefficient $\beta_{-4N}$

Let $E$ be an elliptic curve with conductor $N_E$, and let $\beta_{D}$ denote the coefficient associated with a fundamental discriminant or its square multiple as in the paper. Square-index parity conjecture. If $N_E\equiv 3\pmod{4}$, then the integer $\beta_{-4N}$ is even.

This is presented as a consequence of the preceding conjecture together with the Gross–Kohnen–Zagier relation for $p=2$ and $D=-N$. The claim remains open in the supplied text.