It is known (cf., [Tod80], [Looi81], [KT87]) that the moduli space of all Einstein metrics on a Kähler K3 surfaces (including orbifold-metrics) has again a structure of the locally Riemannian symmetric space:

$$O(\Lambda_{\rm K3})\backslash SO_{0}(3,19)/(SO(3)\times SO(19)),$$

which we denote by $\mathcal{M}_{\rm K3}$. An enriched version encoding also complex structures of the K3 surfaces is

$$\mathbb{R}_{>0}\times(O(\Lambda_{\rm K3})\backslash SO_{0}(3,19)/(SO(2)\times SO(19))).$$

Roughly speaking, this is a union of Kähler cones of ADE K3 surfaces with marking of the minimal resolutions.

Thus we can again compare a Satake compactification of $\mathcal{M}_{\rm K3}$ with the Gromov-Hausdorff compactification. Inside the Satake compactification for the adjoint representation, we consider an open locus (a partial compactification of $\mathcal{M}_{\rm K3}$) $\mathcal{M}_{\rm K3}\sqcup\mathcal{M}_{\rm K3}(a),$ where $\mathcal{M}_{\rm K3}(a)$ denotes the $36$-dimensional boundary stratum corresponding to an isotropic rational line $l=\mathbb{Q}e$ in $\Lambda_{\rm K3}\otimes\mathbb{Q}$, with primitive integral generator $e$, which are unique up to $O(\Lambda_{\rm K3})$. Then for each point $p=\langle e,v_{1},v_{2}\rangle$ in strata $\mathcal{M}_{\rm K3}(a)$, we consider the marked (possibly ADE) K3 surface $X_{p}$ with period $\langle v_{1},v_{2}\rangle$. Then it is known that there is an elliptic K3 surface structure on $X_{p}$ with the fiber class $e$. Then we define $\Phi(p)$ as its base biholomorphic to $\mathbb{P}^{1}$ with the McLean metric, which only depends on $\langle v_{1},v_{2}\rangle$. Similarly to the projective case Theorem 4.4, [OO] proves that for non-algebraic situation:

In [OO], we further explicitly define an extension to the whole Satake compactification $\Phi\colon\overline{\mathcal{M}_{\rm K3}}^{\rm Sat}\to{\it CMet}_{1}$, and conjecture that this is still continuous with respect to the Gromov-Hausdorff topology. For the boundary strata other than $\mathcal{M}_{\rm K3}(a)$, we assign flat tori $\mathbb{R}^{i}/\mathbb{Z}^{i}\ (i=1,2,3)$ modulo $(-1)$-multiplication. We show that $\Phi$ restricted to the closure of the locus which parametrizes $\mathbb{R}^{4}/\mathbb{Z}^{4}$ modulo $\pm 1$, that includes those boundary strata, is continuous. Furthermore, we also prove the restriction of $\Phi$ to the closure of $\mathcal{M}_{\rm K3}(a)$ is continuous by using Weierstrass models.