Logarithmic injectivity-radius conjecture for random mapping tori

Let $S$ be a surface, and let $N$ denote the length of a random monodromy word defining a mapping torus of $S$. The injectivity radius is the radius of the largest embedded metric ball at a point, minimized over the manifold. Logarithmic injectivity-radius conjecture. The injectivity radius of a random mapping torus of $S$ decays as $1/\log^2 N$. The estimate follows for the generating set of $\operatorname{SL}(2,\mathbb{Z})$ discussed in the paper from maximal runs in random words; its validity for general generating sets is left as a conjecture.