Negative surface curvature implies hyperbolicity

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

Negative surface curvature implies hyperbolicity. If $\sigma_+(X)<0$ then $\pi_1(X)$ is word-hyperbolic.

The conjecture is presented as a common generalisation of Greendlinger's lemma and the B.B. Newman spelling theorem. The source notes unresolved issues for non-positively curved Euclidean 2-complexes and one-relator presentation complexes; no resolution is supplied.