Non-positive surface curvature implies asphericity

Let $X$ be a finite, standard, irreducible 2-complex, and let $\sigma_+(X)$ denote its positive surface curvature invariant.

Non-positive surface curvature implies asphericity. If $\sigma_+(X)\leq 0$ then $X$ is aspherical.

The conjecture extends the known asphericity consequence of non-positive irreducible curvature to non-positive surface curvature. The source explains that the key challenge is improving a spherical morphism to an essential map; no resolution is given.