Odaka's CM minimization conjecture for polarized families

Let $C$ be a smooth projective curve, and let $\pi:(X,L)\to C$ be a polarized family whose fiber $(X_0,L_0)$ is K-semistable. Write $\mathrm{CM}((X,L)/C)$ for its CM degree. For another polarized family $\pi':(X',L')\to C$, suppose there is a $C^{\circ}$-isomorphism

$$f^{\circ}:(X,L)\times_C C^{\circ}\cong (X',L')\times_C C^{\circ}.$$

Odaka's CM minimization conjecture. One has

$$\mathrm{CM}((X,L)/C)\leq \mathrm{CM}((X',L')/C).$$

Furthermore, if $(X_0,L_0)$ is K-stable and $X'$ is normal, equality holds if and only if $f^{\circ}$ extends to an isomorphism $f:(X,L)\cong(X',L')$ over all of $C$. This conjecture concerns the minimization of CM degree among models with the same generic fiber; it is known in the Calabi–Yau case, the K-ample case, and the K-(semi)stable log Fano case, while the general statement remains open.