Geometric realization conjecture for higher-dimensional hyperkähler moduli

Fix a connected moduli space $M$ of polarized $2n$-dimensional irreducible holomorphic symplectic manifolds $(X,L)$ with second-cohomology lattice $\Lambda$, and write the associated orthogonal-type locally symmetric space as $\Gamma\backslash\mathcal{D}_{M}$. Let $\overline{\Gamma\backslash\mathcal{D}_{M}}^{\rm Sat,\tau_{ad}}$ be its Satake compactification for the adjoint representation, and let the Gromov–Hausdorff compactification of $M$ be the target. Geometric realization conjecture. There is a continuous map $\Psi$ from $(M\subset)\overline{\Gamma\backslash\mathcal{D}_{M}}^{\rm Sat,\tau_{ad}}$ to the Gromov–Hausdorff compactification of $M$, extending the identity on $M$, such that the $(b_{2}(X)-4)$-dimensional boundary strata parametrize via $\Psi$ the projective space $\mathbb{P}^{n}$ with special Kähler metrics, while the metric spaces parametrized by $0$-dimensional cusps are all homeomorphic to the closed ball of dimension $n$. This is proposed as a higher-dimensional extension of the K3 picture; the supplied text gives no resolution status.