Non-positive curvature for geometric complexes

Let $X$ be a finite, irreducible, geometric 2-complex. Write $\partial M(X)$ for the boundary of its associated 3-manifold and $\pi_1(X)$ for its fundamental group.

Non-positive curvature for geometric complexes. If $\sigma_+(X)>0$ then either $\partial M(X)$ has a spherical component or $\pi_1(X)$ splits freely.

The conjecture is presented as a geometric-complex analogue of Papakyriakopoulos' sphere theorem and is intended to provide surface-curvature bounds relevant to 3-manifold groups; the source gives no resolution.