Minimality conjecture for the pole-order integer in separated variables

Let $p$ be the plane algebraic curve under consideration, let $r$ be the relevant polynomial or function, and suppose $\mathrm{F}(p)$ is trivial. Let $\omega$ be the valuation used in the source, let $(f_\omega,g_\omega)$ be a generator of $\mathrm{F}(\mathrm{lp}_\omega(p))$, and let $k$ be the integer occurring in equation (mult), which relates the leading parts of $f-g$ to $f_\omega^k-g_\omega^k$.

Minimality conjecture. If $\mathrm{F}(p)$ is trivial, then the integer $k$ in equation (mult) is minimal such that

$$\omega(r)<k\cdot\omega(f_\omega).$$

The conjecture is intended to control the multiplicities of poles of separated-variable functions on finite orbits, in a situation where the corresponding upper-bound proposition is unavailable because $\mathrm{F}(p)$ is trivial. Its status is not established in the supplied text.