Boundedness of log-Ding and log-Mabuchi functionals under special degeneration

Let $\pi: ({\mathcal{X}}, {\mathcal{D}})\rightarrow \mathbb C$ be a special degeneration for $(X,D)$. Suppose the central fiber $({\mathcal{X}}_0,{\mathcal{D}}_0)$ admits a singular Kähler–Einstein metric of cone angle $2\pi\beta$ along ${\mathcal{D}}_0$. Functional boundedness conjecture. The log-Ding functional $F_{\omega,(1-\beta)D}$ is bounded below. Consequently, the log-Mabuchi functional ${\mathcal{M}}_{\omega,(1-\beta)D}$ is also bounded below. The statement is motivated by subharmonicity and geodesic convexity of the Ding functional in singular settings; the supplied text gives no resolution status.