Conjecture on tt* structures for quasi-homogeneous Calabi–Yau orbifolds

Let $f$ be a quasi-homogeneous polynomial determining a Calabi–Yau orbifold, and let the monodromy-invariant part of the corresponding Landau–Ginzburg model carry its Landau–Ginzburg $tt^*$ structure. Conjecture on $tt^*$ structures for quasi-homogeneous Calabi–Yau orbifolds. There is some kind of deformation theory of complex structure and Hodge structure on the Calabi–Yau orbifold determined by $f$ that gives a $tt^*$ structure, and this $tt^*$ structure corresponds to the $tt^*$ structure on the monodromy-invariant part of the Landau–Ginzburg side. The claim is explicitly described as vague because the required deformation theory and Hodge theory for the orbifold case have not been developed sufficiently in the source.