Conjectural tt* morphism for the LG/CY correspondence

Let $r^{\prime}:H^{\operatorname{LG}}\to H^{\operatorname{CY}}$ be the correspondence map, and let

$${\mathscr E}^{\operatorname{LG}}=(H^{\operatorname{LG}}\to M,\kappa^{\operatorname{LG}},g^{\operatorname{LG}},D^{\operatorname{LG}},C^{\operatorname{LG}})$$

and

$${\mathscr E}^{\operatorname{CY}}=(H^{\operatorname{CY}}\to M,\kappa^{\operatorname{CY}},g^{\operatorname{CY}},D^{\operatorname{CY}},C^{\operatorname{CY}})$$

be the Landau–Ginzburg and Calabi–Yau $tt^*$ structures. Conjectural $tt^*$ morphism. The map $r^{\prime}:H^{\operatorname{LG}}\to H^{\operatorname{CY}}$ is a morphism of $tt^*$ geometries, meaning that it maps ${\mathscr E}^{\operatorname{LG}}$ to a constant multiple of ${\mathscr E}^{\operatorname{CY}}$. The source presents this as an immediate consequence if the preceding real-structure conjecture holds; it gives no evidence that the resulting morphism statement has been proved independently.