Irreducible curvature bounds of few-relator complexes

Let $X$ be a random few-relator complex with $n$ generators and $m$ relators of length $l$.

Irreducible curvature bounds of few-relator complexes. The probability that $\rho_+(X)=\kappa(X)$ tends to $1$ as $l\to\infty$. In particular, with high probability, $\rho_+(X)<0$ if $m<n-1$ and $\rho_+(X)=0$ if $m=n-1$.

This is the most optimistic proposed estimate for the irreducible curvatures of random few-relator complexes; its status is not resolved in the source.