GGLR's Moishezon conjecture for the period-map Stein factor

Let $D$ be a Mumford--Tate domain parameterizing pure, effective, weight $\mathsf{w}$, $Q$-polarized Hodge structures on a finite-dimensional rational vector space $V$. Let $\Phi:B\to\Gamma\backslash D$ be a period map from a smooth quasi-projective variety $B$, let $\overline B\supset B$ be a smooth projective completion with simple normal crossing boundary, and let $\Phi{}^\mathsf{S}:\overline B\to\overline P$ be the proper topological Satake--Baily--Borel type completion. Write

$$\overline B\xrightarrow{\hat\Phi{}^\mathsf{S}}\hat\wp\longrightarrow\overline P$$

for its Stein factorization. GGLR's Moishezon conjecture. The topological space $\hat\wp$ is Moishezon, and the map $\hat\Phi{}^\mathsf{S}:\overline B\to\hat\wp$ is a morphism. This is known when $D$ is hermitian symmetric and when $\dim\wp\leq 2$. In general, proving the conjecture has been reduced to establishing suitable extension of holomorphic functions near the compact connected fibres of $\hat\Phi{}^\mathsf{S}$; it remains open in the stated generality.