Arapura–Wang's non-negative Euler characteristic conjecture for perverse sheaves

Let $X$ be an aspherical compact Kähler manifold, meaning that its universal cover is contractible, and let ${\mathcal P}$ be a perverse sheaf on $X$. Write $\chi(X,{\mathcal P})$ for the Euler characteristic of ${\mathcal P}$.

Arapura–Wang's conjecture.

$$\chi(X,{\mathcal P})\geq 0.$$

This strengthens the Singer–Hopf conjecture in the Kähler setting. The conjecture is proved when $X$ has non-positive sectional curvature, and also when $X$ is projective and possesses a faithful semisimple rigid local system; it remains open in general.