Perverse-sheaf Singer–Hopf conjecture for aspherical projective manifolds

Let $X$ be an aspherical projective manifold and let $\mathcal{P}$ be a perverse sheaf on $X$. Perverse-sheaf Singer–Hopf conjecture. The Euler characteristic of $\mathcal{P}$ is semipositive:

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

This is proposed as a projective generalization of the Singer–Hopf conjecture and is presented as open in the source.