Existence of nice nef partitions and toric Landau–Ginzburg models for weighted Fano complete intersections

Let $X$ be a smooth well formed Fano weighted complete intersection. A nice nef partition conjecture asserts that there exists a nice nef partition for $X$, and that $X$ has a toric Landau--Ginzburg model.

The preceding results establish this assertion for smooth well formed Fano weighted hypersurfaces and for smooth well formed Fano weighted complete intersections of codimension $2$. The conjecture would extend the existence of toric Landau--Ginzburg models to all smooth well formed Fano weighted complete intersections.