Logarithmic action conjecture for polynomial components of almost Belyi maps

Let $\varphi(x,w)$ be an almost Belyi map with definition field $K=\mathbb{Q}(w)$. Suppose the vector field

$$A(x,w)\frac{\partial}{\partial x}+B(x,w)\frac{\partial}{\partial w}$$

annihilates $\varphi(x,w)$. A logarithmic action conjecture. This vector field acts logarithmically on every polynomial component of $\varphi$, where a polynomial component is any $K[x]$-irreducible factor of a numerator or denominator of $\varphi$ or $\varphi-1$. The conjecture has been checked for all known almost Belyi maps, but no general proof is given; it would imply that $B(x,w)$ is linear in $x$ and constrain the degree of $A(x,w)$.