Separating/nonseparating algebraicity principle for exterior Cauchy transforms

Let $\Gamma$ be a smooth algebraic oval and let $\gamma$ be its lift to the normalization $X$ of the relevant irreducible Schwarz component. The exterior Cauchy transform is expressed through the Abelian-period component of

$$\frac{1}{2\pi i}\int_\gamma\frac{\eta\,d\pi}{z-\pi}.$$

Separating/nonseparating algebraicity principle. If $\gamma$ is separating, then the exterior Cauchy transform is algebraic. If $\gamma$ is nonseparating, then the exterior Cauchy transform is expected to be generically non-algebraic; algebraicity can occur only when the Abelian-period component vanishes identically or cancels with the residue terms in an exceptional way.

This principle relates algebraicity to the topology of the lifted oval and to cancellation in Abelian periods. The statement is presented as a working conjecture, and the additional analytic step converting non-invisibility of periods into non-algebraicity remains open.