Existence of Lefschetz operators for general fans

Determine whether every fan admits a Lefschetz operator acting on its combinatorial intersection cohomology, without assuming that the fan admits a strictly convex piecewise linear function or is projective.

Background

For a fan associated with a toric variety, a Lefschetz operator acts on intersection cohomology. In the combinatorial setting, a strictly convex piecewise linear function provides such an operator when the fan is projective; the resulting operator satisfies the hard Lefschetz theorem. The paper notes that the existence of an analogous operator is unresolved for general fans that may not admit such a function.