Hard Lefschetz theorem and Hodge-Riemann relations for convex valuations

The algebra of smooth translation-invariant valuations on convex bodies, introduced by S.Alesker in the early 2000s, was in part proved and in part conjectured to satisfy properties formally analogous to those of the cohomology ring of a compact Kähler manifold: Poincaré duality, the hard Lefschetz