NMSSM with Gravitino Dark Matter to be tested at LHC

We present a solution to the gravitino problem, which arises in the NMSSM, allowing for sparticle spectra from ordinary gravity-mediated supersymmetry breaking with weak-scale gravitino dark matter. The coupling, which links the singlet to the MSSM sector, enhances the tree-level Higgs mass, providing an attractive explanation why the observed Higgs boson is so heavy. The same coupling induces very efficient pair-annihilation processes of the neutralino NLSP. Its relic abundance can be sufficiently suppressed to satisfy the strong constraints on late decaying relics from primordial nucleosynthesis -- even for very long neutralino lifetimes. The striking prediction of this scenario is the detection of a pseudoscalar Higgs boson in the search for top-top resonances at LHC-14, rendering it completely testable.