The relative $p$-parity conjecture

Let $F$ be a number field, let $L$ be a $p$-adic local field, and let $V$ and $V'$ be geometric symplectic self-dual $p$-adic representations of $G_F$ over $L$. For each representation define

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

Let $\varepsilon(V)$ and $\varepsilon(V')$ denote their global epsilon constants. The relative $p$-parity conjecture.

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

This compares the parity difference of two Bloch--Kato Selmer-group Euler characteristics with the ratio of their global epsilon constants. Its status is not resolved in the supplied text.