The $p$-parity conjecture for geometric symplectic self-dual representations

Let $F$ be a number field, let $L$ be a $p$-adic local field, and let $V$ be a geometric $p$-adic representation of $G_F$ over $L$, meaning that it is unramified outside finitely many places and de Rham at places above $p$. Define

$$\chi_{\rm f}(F,V)=\dim_L H^1_{\rm f}(F,V)-\dim_L H^0(F,V),$$

and let $\varepsilon(V)\in\{\pm1\}$ be its global epsilon constant. The $p$-parity conjecture. If $V$ is symplectic self-dual, then

$$\varepsilon(V)=(-1)^{\chi_{\rm f}(F,V)}.$$

This is a mod-$2$ analogue of the Bloch--Kato conjecture and relates the global epsilon constant to the parity of the Bloch--Kato Selmer-group Euler characteristic. Its status is not resolved in the supplied text.