Following the 2022 alert of a TESS object of interest transiting TOI-5624 (a G7, Vstar ~100pc away), a CHEOPS campaign in 2023 detected four planetary signals at P_b_~3.4, P_c_~7.9, P_d_~13.7, and P_e_~21.5 days, later confirmed by additional TESS and CHEOPS photometry in 2024-2025. By using TESS and CHEOPS photometry along with HARPS-N and SOPHIE high-resolution spectra, we determined the planet properties and performed a dynamical analysis of the system. After analysing the photometric data, we extracted and modelled the radial velocity (RV) time series using two independent methodologies both within a Markov chain Monte Carlo framework. We further integrated the N-body equations of motion, while simultaneously fitting the transit times and the detrended RVs, to dynamically characterise the system. We present the discovery of four transiting sub-Neptunes with radii of R_b_=2.314+/-0.035R_{Earth}_, R_c_=2.474+/-0.042R_{Earth}_, R_d_=3.584_-0.050_^+0.051^ R_{Earth}_, and R_e_=3.247_-0.043_^+0.042^ R_{Earth}_ and masses of M_b_=9.4+/-1.4M_{Earth}_, M_c_=4.8+/-1.9M_{Earth}, M_d_=4.9+/-2.2M_{Earth}_, and M_e_=8.9_-3.0_^+2.9^M_{Earth}_. Our photometric analysis reveals that the outermost transiting planet TOI-5624e shows significant transit time variations (TTVs). Indeed, we find a robust Keplerian signal in the RV time series close to the 2:1 period commensurability with TOI-5624e, which explains the TTV pattern exhibited by TOI-5624e according to our dynamical analysis. We label this non-transiting planet as TOI-5624f and find its minimum mass to be M_f_*sini_f_=13.0+/-3.7M_{Earth}_. Among the known systems hosting more than four planets, the remarkable precision with which the radii have been measured (<1.7%) and the firm assessment (>3{sigma}) of the mass for at least three planets has been previously reached only for TRAPPIST-1. Additional photometric observations will enable a better sample of the TTV modulation and a more robust dynamical determination of the masses.