Random-walk tunnel number one 3-manifolds do not fiber

Let $H$ be the handlebody used to construct tunnel number one 3-manifolds, let $\partial H$ be its boundary, and let $\mathcal{MCG}(\partial H)$ be its mapping class group. Choose a finite generating set and form a random walk of length $r$, applying its endpoint to a fixed non-separating simple closed curve to obtain the attaching curve and the corresponding manifold $M$. Random-walk non-fibering conjecture. The probability that $M$ fibers over the circle goes to $0$ as $r\to\infty$. This is a conjectural analogue of the proved result for the measured-lamination model; the paper presents experimental evidence, but no proof is supplied for the mapping-class-group model.