  Convergence to spatially homogeneous steady states is shown for a chemotaxis
model with local sensing and possibly nonlinear diffusion when the intrinsic
diffusion rate $\phi$ dominates the inverse of the chemotactic motility
function $\gamma$, in the sense that $(\phi\gamma)'\ge 0$. This result
encompasses and complies with the analysis and numerical simulations performed
in Choi \& Kim (2023). The proof involves two steps: first, a Liapunov
functional is constructed when $\phi\gamma$ is non-decreasing. The convergence
proof relies on a detailed study of the dissipation of the Liapunov functional
and requires additional technical assumptions on $\phi$ and $\gamma$.
