Universal covering that induces zero on homologies

Question

Let $p:\tilde{X}\rightarrow X$ be the universal covering space such that $p_*$ is zero on all homologies of dimension greater than zero. Does this imply that $X$ is $K(\pi_1(X),1)$? Working with the second homology groups implies that Hurewicz homomorphism $\pi_2(X)\rightarrow H_2(X)$ is zero while $\pi_2(\tilde{X})\rightarrow H_2(\tilde{X})$ is an isomorphism. I cannot derive any contradictions from here. So maybe it is not true.

Answer 1

A counter-example: $\mathbb{RP}^n$