Continuity of the geometric realization map for polarized K3 surfaces

Let $\mathcal{F}_{2d}$ be the moduli space of polarized K3 surfaces of degree $2d$, and let $\overline{\mathcal{F}_{2d}}^{{\rm Sat},\tau_{\rm ad}}$ be its Satake compactification for the adjoint representation. Let

$$\Phi_{\rm alg}\colon \overline{\mathcal{F}_{2d}}^{{\rm Sat},\tau_{\rm ad}} \to \{\text{compact metric spaces with diameter $1$}\}$$

be the geometric realization map, with the target endowed with the Gromov–Hausdorff topology. The geometric realization conjecture for polarized K3 surfaces. The map $\Phi_{\rm alg}$ is continuous. This predicts that the metric spaces assigned to interior and boundary points of the Satake compactification vary continuously, including the tropical K3 surfaces and segments arising on the boundary; the paper partially confirms the conjecture.