Macroscopic relaxation conjecture for lozenge Glauber dynamics

Let $U$, $\varphi$ and their discretizations $(U_L,\varphi_L)_{L\ge1}$ be as above, and let $h_t^\eta$ denote the height function under Glauber dynamics started from an arbitrary initial condition $\eta$. Let $\bar\phi$ be the equilibrium limit shape and let $L$ be the scaling parameter. Macroscopic relaxation conjecture. For every $\delta>0$ there exists $c(\delta)<\infty$ such that, whatever the initial condition $\eta$, at times $t>c(\delta)L^{2+\delta}$, with probability tending to $1$ as $L\to\infty$, one has for every vertex $v\in U_L$ that

$$|h_t^\eta(v)-\bar\phi(v)|=o(1).$$

Thus, within time $L^{2+o(1)}$, the interface is expected to macroscopically approximate the equilibrium shape to any pre-assigned precision. The conjecture concerns relaxation to the limit shape rather than full convergence to equilibrium; the source does not report a proof or disproof.