Griffiths conjecture on period images

Let $U$ be a quasi-projective variety carrying a $\mathbb Z$-PVHS, with period map $\varphi:U\to\Gamma\backslash\mathcal D$. Let $\mathlcal P$ be the period image, and let $L:=\bigotimes_p\operatorname{det}(\mathrm F^pV_{\mathcal O})$ be the Griffiths line bundle.

Griffiths conjecture. The period image $\mathlcal P$ is quasi-projective and $\varphi$ is induced by an algebraic morphism; $L$ is ample on $\mathlcal P$; and there exists a Baily–Borel type compactification $\mathlcal P^{\mathrm{BB}}$ such that

$$\mathlcal P^{\mathrm{BB}}=\operatorname{Proj}$$

of the ring of moderate-growth sections of $L^{\otimes n}$.

The source states that parts (1) and (2) have been proved in BBT-23; it does not state that part (3) has been resolved.