Jian-Shi-Song's convergence conjecture for cscK metrics on manifolds with semi-ample canonical bundle

Let $(M,\omega_0)$ be a compact Kähler manifold with semi-ample canonical bundle $K_M$. Let $c_1(M)=[\operatorname{Ric}(\omega_0)]$ and consider any sequence of cscK metrics in the Kähler class $-c_1(M)+\varepsilon[\omega_0]$, where $\varepsilon>0$ is sufficiently small. Let $g_{\mathrm{can}}$ be the twisted Kähler–Einstein metric on the canonical model $M_{\mathrm{can}}$, and let $\Phi:M\to M_{\mathrm{can}}$ be the canonical map. Jian-Shi-Song's convergence conjecture. Any such sequence converges to $g_{\mathrm{can}}$ both globally in Gromov–Hausdorff topology and locally in smooth topology away from the singular fibres of $\Phi$. This conjecture concerns the degeneration of cscK metrics toward the canonical model; the source abstract says that the paper confirms it partially by proving smooth convergence on compact subsets away from a subvariety, while the stated global Gromov–Hausdorff and full local convergence remain unresolved in the supplied text.