Local root-number formula for C2D4 curves

Let $K$ be a local field of characteristic $0$, and let $C/K$ be a C2D4 curve with invariants $\mathcal{P}$, $\Delta$, $w_{\operatorname{Jac} C/K}$, $\lambda_{C/K}$, and $E_{C/K}$ as defined in the source. Assume $\mathcal{P}\neq0$ and $\Delta\neq0$.

Local root-number formula. The local root number of the Jacobian satisfies

$$w_{\operatorname{Jac} C/K}=\lambda_{C/K}\cdot E_{C/K}.$$

This gives an explicit local formula intended to support the parity analysis for abelian surfaces. The supplied context does not establish whether the displayed assertion is conjectural or prove its status.