Polynomial separation conjecture for capacitary holomorphic functions

Let $V^+$ be a complex neighborhood of the compact Riemann surface with boundary $V$, let $O\in V$ be the distinguished point, and let $\phi:V^+\to\mathbb{C}$ be holomorphic with vanishing order $e$ at $O$. Let $\sigma$ be an antiholomorphic involution of $V^+$. Write $\phi^{[e]}(O)$ for the leading $e$-th derivative data of $\phi$ at $O$, and let $\|\phi^{[e]}(O)\|^{\mathrm{cap},\otimes e}_{V,O}$ denote its capacitary norm.

Polynomial separation conjecture. If

$$\log\|\phi^{[e]}(O)\|^{\mathrm{cap},\otimes e}_{V,O}<0$$

and $\phi\circ\sigma=\overline{\phi}$ on $V^+$, then there exists a monic polynomial $p(X)\in\mathbb{R}[X]$ such that

$$\inf_{x\in V}\left|p\left(\frac{1}{\phi(x)}\right)\right|>1.$$

This is the complex-analytic conjecture invoked in the paper's arithmetic theorem; its status is not resolved in the supplied material.