Local Prym parity product formula for genus 2 and 3 curves

Let $C$ be a curve of genus $2$ or $3$ over a number field $K$, with an unramified double cover

$$\pi\colon D\to C$$

and associated Prym variety $\operatorname{Prym}(D/C)$ over $K$. Let $\lambda_{C/K_v,\phi_v}$ denote the local factors associated with the relevant isogeny at each place $v$, and let $w_{\operatorname{Jac} C/K_v}$ and $w_{\operatorname{Prym}(D/C)/K_v}$ be the corresponding root numbers. Local Prym parity product formula. One has

$$\prod_v \lambda_{C/K_v,\phi_v}\cdot w_{\operatorname{Jac} C/K_v}\cdot w_{\operatorname{Prym}(D/C)/K_v}=1.$$

Together with the paper’s local theorem, this formula would imply the parity conjecture for product abelian varieties of the form $\operatorname{Jac}(C)\times\operatorname{Prym}(D/C)$. Its validity is proposed in the source and remains open there.