General Lefschetz-operation conjecture for the intersection cohomology of complete fans

Let $\Delta$ be a complete fan of dimension $n$, and let ${\mathcal A}(B\Delta)$ be its associated graded algebra with elements $L_i\in {\mathcal A}(B\Delta)$ of degree $e_i$, for $i=1,\ldots,n$. A Lefschetz operation is the collection of endomorphisms induced by these degree-$e_i$ elements in the construction of the Lefschetz module structure. General Lefschetz-operation conjecture. If each $L_i$ is a general element of degree $e_i$, then the $L_i$ define a Lefschetz operation on ${\mathcal A}(B\Delta)$. This would give a stronger form of the paper's main theorem and, in particular, imply non-negative integer coefficients for the $cd$-index of $\Delta$; the source does not provide a proof or resolution of this stronger assertion.