Conjecture on the extended period map for I-surfaces

Let $\mathcal{M}$ be the moduli space of I-surfaces, let $\widetilde{\mathcal{M}}$ be its natural blowup along the boundary strata, and let $\Phi\colon \mathcal{M}\to\Gamma\backslash D$ be the period map with image $Z$. Let $\mathcal{S}_{1,1,1}$ denote the codimension-$3$ stratum corresponding to the elliptic ruled case with multiplicities $(1,1,1)$. For a boundary point, let $N_i$ be the logarithms of monodromy around the local boundary components, and let $\operatorname{Im}N_i$ be the associated subspaces. Extended period map conjecture. There exists an analytic space $\overline{Z}$, a partial compactification of $Z$, such that: (i) the boundary points of $\widetilde{\Phi}$ contain the nilpotent orbit of the limiting mixed Hodge structure together with the unordered collection of subspaces $\operatorname{Im}N_i$; and (ii) after replacing $\widetilde{\mathcal{M}}$ by $\widehat{\mathcal{M}}$, obtained by a sequence of toric blowups over boundary strata, the period map extends to a holomorphic map

$$\widehat{\Phi}\colon \widehat{\mathcal{M}}\to\overline{Z}.$$

This conjecture is motivated by compactification theories for period maps and by the described monodromy of the boundary. It predicts a partial compactification recording the limiting mixed Hodge data and resolving the period map after toric modifications; the source does not state whether it has been proved or disproved.