Weakly self-avoiding walk is a model of simple random walk paths that
penalizes self-intersections. On $\mathbb{Z}$, Greven and den Hollander proved
in 1993 that the discrete-time weakly self-avoiding walk has an asymptotically
deterministic escape speed, and they conjectured that this speed should be
strictly increasing in the repelling strength parameter. We study a
continuous-time version of the model, give a different existence proof for the
speed, and prove the speed to be strictly increasing. The proof of monotonicity
is by stochastic dominance.