The 2-isogeny local root-number formula

Let $F$ be a local field of characteristic zero, and let $E/F$ be an elliptic curve with a 2-isogeny $\phi:E\to E'$ over $F$. In the notation of the source, let $a,b\in F$ be the parameters appearing in the chosen 2-isogeny model, and let $(\,\ )_F$ denote the Hilbert symbol. 2-isogeny conjecture.

$$w(E/F)=\sigma_\phi(E/F)\cdot\begin{cases}(a,-b)_F(-2a,a^2-4b)_F,&a\ne0,\\(-2,-b)_F,&a=0. \end{cases}$$

The formula predicts a precise local relation between the root number and the isogeny-local term. The surrounding discussion presents it as a conjectural correction formula, and the source does not supply an unconditional resolution in this span.