Algebraic primitive without finite pole-order hypothesis

Ascertain whether one can always construct an algebraic (n−1)-form η on a principal open U ⊂ P^n meeting a smooth hypersurface X such that ω_A|_U − ∂η is holomorphic, given only the existence of a holomorphic primitive β on U \ X for which ω_A|_U − ∂β is holomorphic, even when β may have essential singularities along X (i.e., without assuming a finite pole-order bound for β).

Background

The paper proves (Theorem \ref{grothendieck}) that, under a finite pole-order bound for β along X, one can assume β is algebraic when ω_A|_U − ∂β is holomorphic. This algebraicity is used to derive a necessary and sufficient algebraic PDE criterion for the vanishing sought by the generalized Hodge conjecture in the studied setting.

However, the authors note that the finite pole-order assumption seems essential in their argument; without it (i.e., if β may have essential singularities near X), they do not know how to produce an algebraic primitive analogous to that guaranteed under the pole-order bound.