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 $\approx δ^{-2}$ in any dimension, with $δ\to0$ 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 $δ\to0$ around a deterministic profile $φ$, the unique minimizer of a surface tension functional. Despite some partial mathematical results, the conjecture $T_{mix}=δ^{-2+o(1)}$ has been proven, so far, only in the situation where $φ$ is an affine function. In this work, we prove the conjecture under the sole assumption that the limit shape $φ$ contains no frozen regions (facets).