Negative irreducible curvature implies locally quasiconvex

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

Negative irreducible curvature implies locally quasiconvex. If $\rho_+(X)<0$ then $\pi_1(X)$ is a locally quasiconvex hyperbolic group.

The source notes that Linton proved hyperbolicity for one-relator groups with negative immersions, confirming part of this conjecture in that case; the general statement remains unresolved.