Barannikov–Kontsevich Frobenius submanifold conjecture for Landau–Ginzburg orbifolds

Let $F$ be an LG system over a frame $(Z,X,S,T)$, and let a primitive form $\zeta^{(0)}$ be given. Assume that $r=\sum_{i=0}^n w_i=1$ and that the hypersurface $V$ in the weighted projective space $\mathbb{P}(w_0,\dots,w_n)$ is a smooth Calabi–Yau manifold. Define

$$S':=\{t\in S\mid t^i=0\text{ for }N(\frac{\partial}{\partial t^i})\notin\mathbb{Z}\}.$$

Barannikov–Kontsevich conjecture. The Frobenius structure $(S',\eta_F|_{S'},\Phi_F|_{S'})$ should be a Frobenius submanifold of the Frobenius manifold constructed by Barannikov–Kontsevich for the Calabi–Yau manifold $V$.

This conjecture compares the integral-exponents, or untwisted, sector of the Landau–Ginzburg orbifold theory with the Frobenius geometry arising from the Calabi–Yau hypersurface. The source gives no resolution or further evidence for the conjecture.