Fibers over infinity of Landau-Ginzburg models

We conjecture that the number of components of the fiber over infinity of Landau–Ginzburg model for a smooth Fano variety $X$ equals the dimension of the anticanonical system of $X$. We verify this conjecture for log Calabi–Yau compactifications of toric Landau–Ginzburg models for smooth Fano threefolds, complete intersections, and some toric varieties.