Generalized Lomb-Scargle periodograms of HARPS-N, SOPHIE, and combined RV residuals (from top to bottom) after subtracting the 1pl model (left), 1pGP model (centre), and 1pGP+FWHM model (right). The vertical red dotted line indicates the period of TOI-1710b at 24.28 d. The shaded orange region indicates the 3σ region of the P_rot from the joint fit (see Sect. 4.3). The 10%, 1%, and 0.1% FAP levels are indicated by the grey dotted, dash-dotted, and dashed lines, respectively.