Norm attaining multilinear forms and polynomials on preduals of Lorentz sequence spaces

For each natural number N, we give an example of a Banach space X such that the set of norm attaining N-linear forms is dense in the space of all continuous N-linear forms on X, but there are continuous (N + 1)-linear forms on X which cannot be approximated by norm attaining (N + 1)-linear forms. Actually, X is the canonical predual of a suitable Lorentz sequence space. We also get the analogous-result for homogeneous polynomials.