Prestellar cores, by definition, are gravitationally bound but starless pockets of dense gas. Physical conditions that could render a core starless (in the local Universe) is the subject of investigation in this work. To this end, we studied the evolution of four starless cores, B68, L694-2, L1517B, L1689, and L1521F, a VeLLO. We demonstrate: (i) cores contracted in quasistatic manner over a timescale on the order of ~ 105 yr. Those that remained starless briefly acquired a centrally concentrated density configuration that mimicked the profile of a unstable BonnorEbert sphere before rebounding, (ii) three cores viz. L694-2, L1689-SMM16, and L1521F remained starless despite becoming thermally super-critical. By contrast, B68 and L1517B remained sub-critical; L1521F collapsed to become a VeLLO only when gas-cooling was enhanced by increasing the size of dust-grains. This result is robust, for other starless cores viz. B68, L694-2, L1517B, and L1689 could also be similarly induced to collapse. The temperature-profile of starless cores and those that collapsed was found to be radically different. While in the former type, only very close to the centre of a core was there any evidence of decline in gas temperature, by contrast, a core of the latter type developed a more uniformly cold interior. Our principle conclusions are: (a) thermal super-criticality of a core is insufficient to ensure it will become protostellar, (b) potential star-forming cores (the VeLLO L1521F here), could be experiencing dust-coagulation that must enhance gasdust coupling and in turn lower gas temperature, thereby assisting collapse. This also suggests, mere gravitational/virial boundedness of a core is insufficient to ensure it will form stars.