Singular Singer–Hopf conjecture for subvarieties

Let $X$ be an aspherical complex projective manifold and let $Z\subseteq X$ be a closed irreducible subvariety. Let $Eu_Z$ be MacPherson's local Euler obstruction function, let $\chi(Z,Eu_Z)$ be the Euler–Mather characteristic, let $\chi^{IH}(Z)$ be the intersection cohomology Euler characteristic, and let $ic_Z$ be the constructible function obtained from the stalkwise Euler characteristic of the intersection-cohomology complex $IC_Z$. Let $\nu_Z$ be Behrend's constructible function and set $\chi_{vir}(Z):=\chi(Z,\nu_Z)$. Singular Singer–Hopf conjecture. The following inequalities hold:

$$(-1)^{\dim_\mathbb{C}Z}\cdot\chi(Z,Eu_Z)\geq 0,$$ $$(-1)^{\dim_\mathbb{C}Z}\cdot\chi^{IH}(Z)=\chi(Z,ic_Z)\geq 0,$$ $$\chi_{vir}(Z)\geq 0.$$

When $Z=X$, these statements reduce to the complex projective Singer–Hopf conjecture, and when $Z$ is smooth they reduce to the corresponding signed Euler-characteristic inequality. The paper proposes these singular variants; their general validity remains open.