Uniform non-negativity conjecture for constructible functions

Let $X$ be an aspherical complex projective manifold. A constructible function on $X$ has an effective characteristic cycle when its characteristic cycle is effective. Uniform non-negativity conjecture. If $\varphi$ is a constructible function on $X$ with effective characteristic cycle, then

$$\chi(X,\varphi)\geq 0.$$

The paper states that this conjecture is equivalent to the first singular Singer–Hopf inequality above. Its general validity is not established in the cited text.