The surface lithium abundance A(Li) of warm metal-poor dwarf stars exhibits a narrow plateau down to [Fe/H]~-2.8dex, while at lower metallicities the average value drops by 0.3dex with a significant star-by-star scatter (called lithium meltdown). This behaviour is in conflict with predictions of standard stellar evolution models calculated with the initial A(Li) provided by the standard Big Bang nucleosynthesis. The lower red giant branch (LRGB) stars provide a complementary tool to understand the initial A(Li) distribution in metal-poor stars. We have collected a sample of high-resolution spectra of 58 LRGB stars spanning a range of [Fe/H] between ~-7.0dex and ~-1.3dex. The LRGB stars display an A(Li) distribution clearly different from that of the dwarfs, without signatures of a meltdown and with two distinct components: (a) a thin A(Li) plateau with an average A(Li)=~1.09+/-0.01dex ({sigma}=~0.07dex), and (b) a small fraction of Li-poor stars with A(Li) lower than ~0.7dex. The A(Li) distribution observed in LRGB stars can be reconciled with an initial abundance close to the cosmological value, by including an additional chemical element transport in stellar evolution models. The required efficiency of this transport allows us to match also the Spite plateau lithium abundance measured in the dwarfs. The emerging scenario is that all metal-poor stars formed with the same initial A(Li) but those that are likely the product of coalescence or that experienced binary mass transfer and show lower A(Li). We conclude that A(Li) in LRGB stars is qualitatively compatible with the cosmological A(Li) value and that the meltdown observed in dwarf stars does not reflect a real drop of the abundance at birth.