The twisted parity conjecture for abelian varieties

Let $\mathcal{F}/\mathcal{K}$ be a finite Galois extension, let $A$ be an abelian variety over the global field $\mathcal{K}$, and let $A(\mathcal{F})_{\mathbb{C}}$ be the $\operatorname{Gal}(\mathcal{F}/\mathcal{K})$-representation on $A(\mathcal{F})\otimes_{\mathbb{Z}}\mathbb{C}$. Let $\tau$ be an Artin representation of $\operatorname{Gal}(\mathcal{F}/\mathcal{K})$ with real trace, and let $\langle A(\mathcal{F})_{\mathbb{C}},\tau\rangle$ denote the relevant representation-theoretic multiplicity. Twisted parity conjecture.

$$(-1)^{\langle A(\mathcal{F})_{\mathbb{C}},\tau\rangle}=W(A/\mathcal{K},\tau).$$

This generalizes the parity conjecture to self-dual Artin twists and is motivated by the Birch and Swinnerton-Dyer conjecture for Artin twists. The source presents it as a conjecture and gives no resolution in this generality.