It is pointed out that a bino-dominated well-tempered bino-higgsino in the Minimal Supersymmetric Standard Model (MSSM) with heavy non-SM-like scalars can satisfy the 2016 LUX constraints on the scattering cross-section of dark matter on nuclei only if tan⁡β is smaller than about 3. This, together with the Higgs mass constraint, sets a lower bound on the stops masses of about 25 TeV. The LUX constraints can be satisfied for larger tan⁡β if the non-SM-like Higgs bosons are light enough. However, this region of parameter space is strongly constrained by recent LHC results of the Higgs boson searches. Satisfying both the LUX and LHC constraints requires the non-SM-like Higgs bosons to be lighter than about 400 GeV and tan⁡β below about 8. This implies a lower bound on the stop masses of about 1.5 TeV. This small corner of the parameter space will be probed in the near future by the direct detection experiments, the LHC Higgs searches and precision Higgs coupling measurements. The recent LUX constraints improved also the lower mass limit on higgsino-dominated well-tempered neutralino to about 950 (900) GeV with heavy (light) MSSM-like Higgs doublet, assuming the stop masses below 10 TeV.