The local root-number formula for Sturm-polynomial error terms

Let $\mathcal K$ be a local field of characteristic $0$, and let $f(x)\in\mathcal K[x]$ be separable and monic with $f(0)\neq0$. Define hyperelliptic curves over $\mathcal K$ by

$$C_1:y^2=f(x),\qquad C_2:y^2=xf(x).$$

Let $P_0,P_1,\ldots$ be the Sturm sequence for $f$. Assume that $\deg P_i=\deg f-i$ for $i=0,\ldots,\deg f$ and that $\prod_{i=0}^{\deg f-1}P_i(0)\neq0$.

Local conjecture. Under these hypotheses,

$$w_{\operatorname{Jac} C_1/\mathcal K}w_{\operatorname{Jac} C_2/\mathcal K}=\lambda_{f,\mathcal K}H_{f,\mathcal K}.$$

This claim is presented as a conjecture based on experimental data and concerns a local formula relating Jacobian root numbers to the factors $\lambda_{f,\mathcal K}$ and $H_{f,\mathcal K}$. The supplied text does not state whether it has since been proved or disproved.