Nef-partition conjecture for smooth well-formed weighted complete intersections

Let $X$ be a smooth well-formed weighted complete intersection. A nef-partition is the toric partition required in the generalized Givental construction, and a good toric Landau–Ginzburg model is the corresponding Laurent-polynomial mirror model.

Nef-partition conjecture. The variety $X$ has a good nef-partition and a toric Landau–Ginzburg model.

The source explains that existence of nef-partitions for weighted complete intersections is expected but not proved in general. Thus the asserted existence remains open in the stated generality.