Log-Sobolev inequality for 2D $\mathrm{SU}(N)$ lattice gauge theory at all couplings

We prove that the log-Sobolev inequality holds for the two-dimensional $SU(N)$ Wilson lattice Yang--Mills measure with arbitrary fixed inverse coupling $β$. The log-Sobolev constant depends on $β$ and $N$, but is uniform in the volume of the underlying lattice. The proof relies on a two-scale disintegration argument.