The derivation of the state of the art tensorial versions of Fundamental Measure Theory (a form of classical Density Functional Theory for hard spheres) is reexamined in the light of the recently introduced concept of global stability of the density functional based on its boundedness [Lutsko and Lam, Phys. Rev. E 98, 012604 (2018)]. It is shown that within the present paradigm, explicit stability of the functional can be achieved only at the cost of giving up accuracy at low densities. It is argued that this is an acceptable trade-off since the main value of DFT lies in the study of dense systems. Explicit calculations for a wide variety of systems show that a proposed explicitly stable functional is competitive in all ways with the popular White Bear models while sharing some of their weaknesses when applied to non-close-packed solids.