The unique coherent extension conjecture for balanced dessins

Let $\tilde{\mathcal{A}}$ be the algebra generated by balanced dessins, and for each $\psi\in\tilde{\mathcal{A}}$ let $P_\psi(x)$ be its minimal polynomial. A partial choice of prime factors $Z_\psi(x)$ is obtained for the elements $\psi=\psi_D$ associated with dessins $D\in\mathcal{D}$, where the selected factor is required to satisfy

$$\deg Z_\psi=|\mathcal{G}\psi|.$$

Unique coherent extension conjecture. This partial choice of factors has a unique coherent extension to $\tilde{\mathcal{A}}$, meaning that the ideal generated by $Z_\psi(\psi)$ for the extended choice gives a quotient

$$\tilde{\mathcal{A}}/\mathcal{Z}$$

that is a field. Such an extension would provide the proposed field quotient associated with the algebra of balanced dessins.