Hard Lefschetz conjecture for combinatorial intersection cohomology of arbitrary polytopes

Let $\Delta\subset\mathbb{R}^d$ be a polytope, let $\Phi$ be its dual fan, and let $\overline{M}_\Phi$ be the quotient combinatorial intersection cohomology module defined from the basic sheaf on $\Phi$. For a linear functional $\xi\in\mathbb{R}^{d*}$, let $S_\Delta(\xi)$ be its maximum on $\Delta$; this gives a piecewise-linear element $S_\Delta\in O_\Phi$. Hard Lefschetz conjecture. For every $k<d/2$, multiplication by $S_\Delta^{d-2k}$ should establish an isomorphism

$$\overline{M}^k_\Phi\longrightarrow\overline{M}^{d-k}_\Phi,$$

and multiplication by $S_\Delta$ should be an embedding

$$\overline{M}^k_\Phi\longrightarrow\overline{M}^{k+1}_\Phi.$$

This is the combinatorial analogue of the Hard Lefschetz theorem for toric varieties. It is known for integral polytopes, while the source states that it is believed to hold for arbitrary polytopes.