Non-vanishing conjecture for Mellin transformations

Let $U\subset X$ be complex manifolds satisfying the conditions in Theorem. Let $\mathcal F\in D^b_c(X,A)$ be a constructible complex such that $\mathcal F|_U\neq 0$. Non-vanishing conjecture. Then

$$\mathfrak M_*(\mathcal F|_U)\neq 0.$$

This conjecture extends the non-vanishing result known for certain projective aspherical manifolds. It is proved when $U$ is algebraic and admits a quasi-finite map to a semi-abelian variety, including complements of essential linear hyperplane arrangements, toric arrangements, and elliptic arrangements; it remains open in the general setting.