  Convergence to a steady state in the long term limit is established for
global weak solutions to a chemotaxis model with degenerate local sensing and
consumption, when the motility function is C^1-smooth on [0, $\infty$),
vanishes at zero, and is positive on (0, $\infty$). A condition excluding that
the large time limit is spatially homogeneous is also provided. These results
extend previous ones derived for motility functions vanishing algebraically at
zero and rely on a completely different approach.
