Since thin disc stars are younger than thick disc stars on average, the thin disc is predicted by some models to start forming after the thick disc had formed, around 10 Gyr ago. Accordingly, no significant old thin disc population should exist. Using 6-D coordinates from Gaia-DR2 and age estimates from Sanders & Das (2018), we select $sim 24000$ old stars (${tau > 10}$ Gyr, with uncertainties $lesssim 15%$) within 2 kpc from the Sun (full sample). A cross-match with APOGEE-DR16 ($sim 1000$ stars) reveals comparable fractions of old chemically defined thin/thick disc stars. We show that the full sample pericenter radius ($r_mathrm{per}$) distribution has three peaks, one associated with the stellar halo and the other two having contributions from the thin/thick discs. Using a high-resolution $N$-body+Smooth Particle Hydrodynamics simulation, we demonstrate that one peak, at $r_mathrm{per}approx 7.1$ kpc, is produced by stars from both discs which were born in the inner Galaxy and migrated to the Solar Neighbourhood. In the Solar Neighbourhood, $sim 1/2$ ($sim 1/3$) of the old thin (thick) disc stars are classified as migrators. Our results suggest that thin/thick discs are affected differently by radial migration inasmuch as they have different eccentricity distributions, regardless of vertical scale heights. We interpret the existence of a significant old thin disc population as evidence for an early co-formation of thin/thick discs, arguing that clump instabilities in the early disc offer a compelling explanation for the observed trends.
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