Period-map conjecture for lattice-polarized weak del Pezzo pairs

Fix a smooth elliptic curve $C$. Let $(X,C,\mu)$ be a marked $L$-polarized weak del Pezzo pair of degree $d$ with anticanonical curve $C$. The period target is

$$\operatorname{Hom}(f_d^{\perp}/R_L,C),$$

where $R_L$ is the root sublattice associated to the lattice polarization $L$.

Period-map conjecture. There is a period map from the moduli space of such marked pairs to $\operatorname{Hom}(f_d^{\perp}/R_L,C)$, and this period map is an isomorphism over a dense open set $U\subset\operatorname{Hom}(f_d^{\perp}/R_L,C)$.

This is a global Torelli-type assertion for marked lattice-polarized weak del Pezzo pairs. The source reports partial results on period points and surjectivity, but leaves the stated generic isomorphism conjectural.