Current data (LHC direct searches, Higgs mass, dark matter-related bounds) severely affect the constrained minimal SUSY standard model (CMSSM) with neutralinos as dark matter candidates. But the evidence for neutrino masses coming from oscillations requires extending the SM with at least right-handed neutrinos with a Dirac mass term. In turn, this implies extending the CMSSM with right-handed sneutrino superpartners, a scenario we dub $tilde u$CMSSM. These additional states constitute alternative dark matter candidates of the superWIMP type, produced via the decay of the long-lived next-to-lightest SUSY particle (NLSP). Here we consider the interesting and likely case where the NLSP is a $tilde{tau}$: despite the modest extension with respect to the CMSSM this scenario has the distinctive signatures of heavy, stable charged particles. After taking into account the role played by neutrino mass bounds and the specific cosmological bounds from the big bang nucleosynthesis in selecting the viable parameter space, we discuss the excellent discovery prospects for this model at the future runs of the LHC. We show that it is possible to probe $tilde{tau}$ masses up to 600 GeV at the 14 TeV LHC with $mathcal{L} = 1100$ fb$^{-1}$ when one considers a pair production of staus with two or more hard jets through all SUSY processes. We also show the complementary discovery prospects from a direct $tilde{tau}$ pair production, as well as at the new experiment MoEDAL.
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