CscK metrics near the canonical class

Let $X$ be a Kähler manifold with semi-ample canonical bundle $K_X$. It is proved by Jian-Shi-Song that for any Kähler class $\gamma$, there exists $\delta >0$ such that for all $t\in (0,\delta )$ there exists a unique cscK metric $g_t$ in $K_X+t\gamma$. In this paper, we prove that $\{(X,g_t){\}}_{t\in (0,\delta )}$ have uniformly bounded Kähler potentials, volume forms and diameters. As a consequence, these metric spaces are pre-compact in the Gromov-Hausdorff sense.