Surface curvature of random complexes in the density model

Let $X$ be a random presentation complex in the density model, with density $d$ satisfying $0<d<1/2$.

Surface curvature of random complexes in the density model. The probability that $\sigma_+(X)<0$ tends to $1$ as $l\to\infty$.

The source notes that this would extend the known high-probability $C(7)$ estimate beyond density $d<1/12$ to all densities below $1/2$, but gives no resolution.