The projective compactification conjecture for canonical moduli-part period maps

Let $W_0\subset W$ be the base locus of the relevant smooth family, let $p^c\colon W_0\to D^c/\Gamma$ be the period map of the canonical moduli part, and set

$$P_0=p^c(W_0).$$

Let $\mathcal M^c_W$ denote the canonical moduli part on $W$. The projective compactification conjecture. There exists a projective compactification $P$ of $P_0$ such that: (i) $P$ admits a structure of a normal complex analytic variety; (ii) there exists an analytic extension $p^c_e\colon W\to P$; and (iii) there exists an integer $r$ such that

$$(\mathcal M^c_W)^{\otimes r}=(p^c_e)^*\mathcal O^{an}_{P}(1).$$

The conjecture concerns compactifying the image of the canonical-moduli-part period map and is motivated by analogous compactification results and conjectures for period-map images; the source gives no resolution.