  Global weak solutions to a chemotaxis model with local sensing and
consumption are shown to converge to spatially homogeneous steady states in the
large time limit, when the motility is assumed to be positive and $C^1$-smooth
on $[0,\infty)$. The result is valid in arbitrary space dimension $n\ge 1$ and
extends a previous result which only deals with space dimensions $n\in
\{1,2,3\}$.
