On the existence of nef-partitions for smooth well-formed Fano weighted complete intersections

A nef-partition for a weighted complete intersection is a combinatorial structure on its weights and degrees which is important for Mirror Symmetry. It is known that nef-partitions exist for smooth well-formed Fano weighted complete intersections of small dimension or codimension, and that in these cases they are strong in the sense that they can be realized as fibers of morphisms of weighted simplicial complexes, i.e., finite abstract simplicial complexes equipped with a weight function.
It was conjectured that this approach can be extended to the case of arbitrary codimension. We show that in the case of any codimension greater than $3$ strong nef-partitions may not exist, and provide a sufficient combinatorial condition for existence of a strong nef-partition in terms of weights.
We also show that the combinatorics of smooth well-formed weighted complete intersections can be arbitrarily complicated from the point of view of simplicial geometry.