Toroidal compactification conjecture for elliptic-surface moduli

Let $\overline{F}^R$ be the compactification obtained from stable semi-log-canonical pairs $(\overline S,\epsilon R)$, let $(\overline{F}^R)^\nu$ denote its normalization, and let $\overline{\mathcal D/\Gamma}^{\mathfrak F}$ be a toroidal compactification of the period-domain quotient for a suitable fan $\mathfrak F$.

Toroidal compactification conjecture. There is a morphism

$$(\overline{F}^R)^\nu\to\overline{\mathcal D/\Gamma}^{\mathfrak F}$$

to some toroidal compactification, for an appropriately chosen fan $\mathfrak F$.

The conjecture asks whether the normalization of the KSBA compactification retains enough period information to map to a toroidal compactification. An analogous result is known for elliptic K3 surfaces, but no resolution is supplied here for the elliptic surfaces under consideration.