Hard Lefschetz conjecture for ample classes on complete fans

Let $\Phi$ be a complete fan in a vector space $V$ of dimension $n$. Let $\Omega^1_\Phi$ be the sheaf with stalk $\Omega^1_{\Phi,\sigma}=\operatorname{Span}(\sigma)^\perp$, and let $\overline l\in H^1(\Phi;\Omega^1_\Phi)$ be ample, meaning that it admits a strictly convex lifting $l$. Let $IH(\Phi)$ be the graded intersection cohomology of $\Phi$, and write $IH(\Phi)^{(j)}$ for its degree-$j$ part. Hard Lefschetz conjecture. The class $\overline l$ induces a Lefschetz operator on $IH(\Phi)$: for every $i$, the map

$$\overline l^i:IH(\Phi)^{(n-i)}\to IH(\Phi)^{(n+i)}$$

is an isomorphism. This is the formulation of Hard Lefschetz for arbitrary complete fans, generalizing the classical theorem in the rational projective case. The source develops it as the key conjectural input for the ensuing combinatorial invariance results.