Gromov-Hausdorff Convergence of Constant Scalar Curvature Kähler Metrics near the Canonical Class

Let $X$ be a compact Kähler manifold with semi-ample canonical bundle. For any Kähler class $[ω]$, Jian--Shi--Song proved that there exists a unique cscK metric in the class $2πc_1(K_X)+\varepsilon[ω]$ when $\varepsilon \in (0,\varepsilon_0]$, using the properness criterion of Chen--Cheng. We prove