Continuity of the geometric realization map for Ricci-flat K3 metrics

Let $\mathcal{M}_{\rm K3}$ be the moduli space of Ricci-flat Kähler K3 surfaces modulo hyperKähler rotation, and let $\overline{\mathcal{M}_{\rm K3}}^{\rm Sat,\tau_{\rm ad}}$ be its Satake compactification for the adjoint representation. Let

$$\Phi\colon \overline{\mathcal{M}_{\rm K3}}^{\rm Sat,\tau_{\rm ad}} \to \{\text{compact metric spaces with diameter $1$}\}$$

be the geometric realization map, with the target endowed with the Gromov–Hausdorff topology. The geometric realization conjecture for Ricci-flat K3 metrics. The map $\Phi$ is continuous. This conjecturally identifies the Satake compactification with the Gromov–Hausdorff compactification obtained from collapsing Ricci-flat Kähler K3 metrics; the paper presents the construction and partial evidence but does not establish continuity in full generality.