Quadratic-twist parity conjecture for elliptic curves

Let $E$ be an elliptic curve over $\mathbf Q$ with conductor $N_E$, and let $d$ be a squarefree integer coprime to $2\cdot N_E$. Write $E_d$ for the quadratic twist of $E$ by $d$, and let $\chi_d$ be the quadratic Dirichlet character associated to the field $\mathbf Q(\sqrt d)$. Quadratic-twist parity conjecture.

$$(-1)^{\operatorname{rank} E_d(\mathbf Q)}=(-1)^{\operatorname{rank} E(\mathbf Q)}\cdot\chi_d(-N_E).$$

This is the twist-sensitive version of the parity conjecture used in the paper to obtain evidence for the squarefree-value residue class conjecture in low degrees. It is presented as a conjectural input rather than proved in the source.