  For each smooth Fano threefold $X$ with Picard number 1 we consider a weak
Landau--Ginzburg model, that is a fibration over $\mathbb C^1$ given by a
certain Laurent polynomial. In the spirit of L. Katzarkov's program we prove
that the number of irreducible components of the central fiber of its
compactification is $h^{1,2}(X)+1$. In particular, it does not depend on the
compactification. The question of dependence on the model is open; however we
produce examples of different weak Landau--Ginzburg models for the same variety
with the same number of components of the central fiber.

Algebraic Geometry