Generic non-algebraicity conjecture for smooth Weierstrass cubics

Let $E$ be a smooth real Weierstrass cubic in the stated affine Euclidean model, with compact oval and exterior Cauchy transform obtained from the corresponding differential $\Omega_z$. The reduction of $\Omega_z$ involves elliptic $\zeta$- and $\sigma$-terms together with an algebraic residue part.

Smooth Weierstrass cubic period-test conjecture. For a generic smooth real Weierstrass cubic, the compact oval is nonseparating and the exterior Cauchy transform of the domain bounded by it is not algebraic. Equivalently, the elliptic-period contribution in the reduction of $\Omega_z$ does not cancel algebraically with the residue part.

This gives an explicit elliptic test for the proposed nonseparating obstruction. The source describes the issue as an expected cancellation problem and provides no resolution of the generic assertion.