Multi-oval generic non-algebraicity conjecture for exterior Cauchy transforms

Let $X$ be the normalization of the relevant irreducible real Schwarz component, and suppose that $X(\mathbb{R})$ has at least two smooth connected components. Each individual component is nonseparating. The relevant Abelian-period family is

$$\left\{\eta\,\pi^n d\pi\right\}_{n\ge0}.$$

Multi-oval generic non-algebraicity conjecture. For a generic affine Euclidean realization, the exterior Cauchy transform of the domain bounded by any one of these ovals is non-algebraic. Algebraicity for one oval can occur only if its homology class lies in the annihilator of the Abelian-period part of this family after exact and residue contributions have been removed; this invisibility is not stable under a generic affine realization.

The conjecture allows exceptional examples in which a single nonseparating oval is invisible to the relevant period map. It predicts that robust positive-genus single-oval algebraic situations occur only in special one-real-component dividing models, including certain even-degree hyperelliptic or acnodal plane models.