Compton-y profiles of short (L_f < 9 Mpc), medium-length (9 ≤ L_f < 20 Mpc), and long filaments (L_f ≥ 20 Mpc) in the left, middle, and right panels, respectively. The dark colours show the SZ signal from inter-filament gas, that is gas that excludes the contribution of galactic haloes (massive M_tot > 10¹² M_⊙ haloes hosting at least one galaxy of stellar mass 10⁹ ≤ M_∗ ≤ 10¹² M_⊙). The light colours correspond to the SZ signal in filaments, where the contributions of galactic haloes are included. The lower and upper limits are computed with Eqs. (8) and (9), respectively, using the mean filament lengths of each population: (short), 13.3 (medium), and 27.1 Mpc (long filaments).