Conjecture on semiampleness and ampleness of the extended Hodge line bundle

Let $B$ be the base of a period map $\Phi:B\to\Gamma\backslash D$, let $\overline{B}$ be a completion with boundary divisors $Z_i$, and let $\Lambda_\mathrm{e}$ denote the extended Hodge line bundle on $\overline{B}$. Assume that the differential of $\Phi$ is generically injective. Semiampleness and ampleness conjecture. (a) There are rational numbers $a_i\geq 0$ such that the $\mathbb{Q}$-line bundle

$$\Lambda_\mathrm{e}-\sum a_i[Z_i]$$

is semi-ample. (b) Under suitable local Torelli-type assumptions, there exist integers $a_i\geq 0$ and an integer $m_0$ such that

$$m\Lambda_\mathrm{e}-\sum a_i[Z_i]$$

is ample for every $m\geq m_0$. The conjecture strengthens the expected semiampleness of the Hodge line bundle beyond the classical Hermitian symmetric setting; the local Torelli assumptions give the stronger eventual ampleness statement.