The odd-prime-isogeny local root-number formula

Let $\mathcal K$ be a local field of characteristic zero, and let $E/\mathcal K$ be an elliptic curve with a $p$-isogeny $\phi:E\to E'$ over $\mathcal K$. Put $\mathcal F=\mathcal K(\ker\phi)$, and let $( -1,\mathcal F/\mathcal K)$ denote the relevant quadratic character factor. Odd-prime-isogeny conjecture.

$$w(E/\mathcal K)=\sigma_\phi(E/\mathcal K)(-1,\mathcal F/\mathcal K).$$

This is the proposed local analogue of the 2-isogeny formula for an isogeny of odd prime degree. The source does not establish it in the supplied passage.