  We consider a noninteracting unbounded spin system with conservation of the
mean spin. We derive a uniform logarithmic Sobolev inequality (LSI) provided
the single-site potential is a bounded perturbation of a strictly convex
function. The scaling of the LSI constant is optimal in the system size. The
argument adapts the two-scale approach of Grunewald, Villani, Westdickenberg
and the second author from the quadratic to the general case. Using an
asymmetric Brascamp-Lieb-type inequality for covariances, we reduce the task of
deriving a uniform LSI to the convexification of the coarse-grained
Hamiltonian, which follows from a general local Cram\'{e}r theorem.
