Singer–Hopf conjecture for aspherical closed manifolds

Let $X$ be an aspherical closed manifold with real dimension $2N$.

Singer–Hopf conjecture.

$$(-1)^N\chi(X)\geq 0.$$

This is a sign conjecture for the Euler characteristic of aspherical even-dimensional closed manifolds, arising in the study of $L^2$-type invariants. Its resolution status is not specified in the source.