On the Hard Lefschetz property of stringy Hodge numbers

For projective varieties with a certain class of ‘mild’ isolated singularities and for projective threefolds with arbitrary Gorenstein canonical singularities, we show that the stringy Hodge numbers satisfy the Hard Lefschetz property (i.e. for p+q⩽d−2, where d is the dimension of the variety). This result fits nicely with a 6-dimensional counterexample of Mustaţă and Payne for the Hard Lefschetz property for stringy Hodge numbers in general. We also give such an example, ours is a hypersurface singularity.