Hot planets around cool stars -- two short-period mini-Neptunes transiting the late K-dwarf TOI-1260

We present the discovery and characterization of two sub-Neptunes in close orbits, as well as a tentative outer planet of a similar size, orbiting TOI-1260 - a low metallicity K6V dwarf star. Photometry from TESS yields radii of $R_{\rm b} = 2.33 \pm 0.10$ $R_{\oplus}$ and $R_{\rm c} = 2.82 \pm 0.15$ $R_{\oplus}$, and periods of 3.13 and 7.49 days for TOI-1260b and TOI-1260c, respectively. We combined the TESS data with a series of ground-based follow-up observations to characterize the planetary system. From HARPS-N high-precision radial velocities we obtain $M_{\rm b} = 8.61_{ - 1.46 } ^ { + 1.36 }$ $M_{\oplus}$ and $M_{\rm c} = 11.84_{ - 3.23 } ^ { + 3.38 }$ $M_{\oplus}$. The star is moderately active with a complex activity pattern, which necessitated the use of Gaussian process regression for both the light curve detrending and the radial velocity modelling, in the latter case guided by suitable activity indicators. We successfully disentangle the stellar-induced signal from the planetary signals, underlining the importance and usefulness of the Gaussian Process approach. We test the system's stability against atmospheric photoevaporation and find that the TOI-1260 planets are classic examples of the structure and composition ambiguity typical for the $2-3$ $R_{\oplus}$ range.