  The broad motivation of this work is a rigorous understanding of reversible,
local Markov dynamics of interfaces, and in particular their speed of
convergence to equilibrium, measured via the mixing time $T_{mix}$. In the
$(d+1)$-dimensional setting, $d\ge2$, this is to a large extent mathematically
unexplored territory, especially for discrete interfaces. On the other hand, on
the basis of a mean-curvature motion heuristics and simulations, one expects
convergence to equilibrium to occur on time-scales of order $\approx
\delta^{-2}$ in any dimension, with $\delta\to0$ the lattice mesh.
  We study the single-flip Glauber dynamics for lozenge tilings of a finite
domain of the plane, viewed as $(2+1)$-dimensional surfaces. The stationary
measure is the uniform measure on admissible tilings. At equilibrium, by the
limit shape theorem, the height function concentrates as $\delta\to0$ around a
deterministic profile $\phi$, the unique minimizer of a surface tension
functional. Despite some partial mathematical results, the conjecture
$T_{mix}=\delta^{-2+o(1)}$ has been proven, so far, only in the situation where
$\phi$ is an affine function. In this work, we prove the conjecture under the
sole assumption that the limit shape $\phi$ contains no frozen regions
(facets).
