Context. Following the 2022 alert of a TESS object of interest transiting TOI-5624 (a G7 V star ∼ 100 pc away), a CHEOPS campaign in 2023 detected four planetary signals at Pb ≍ 3.4, Pc ≍ 7.9, Pd ≍ 13.7, and Pe ≍ 21.5 days. These signals were later confirmed by additional TESS and CHEOPS photometry in 2024-2025. Aims. 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. Methods. 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. Results. We present the discovery of four transiting sub-Neptunes with radii of Rb = 2.314 ± 0.035 R⊕, Rc = 2.474 ± 0.042 R⊕, Rd = 3.584−0.050+0.051 R⊕, and Re = 3.247−0.043+0.042 R⊕, along with masses of Mb = 9.4 ± 1.4M⊕, Mc = 4.8 ± 1.9M⊕, Md = 4.9 ± 2.2M⊕, and Me = 8.9−3.0+2.9 M⊕. Our photometric analysis revealed that the outermost transiting planet TOI-5624e shows significant transit-timing variations (TTVs). Indeed, we found a robust Keplerian signal in the RV time series close to the 2:1 period commensurability with TOI-5624 e, which explains the TTV pattern exhibited by TOI-5624 e according to our dynamical analysis. We labelled this non-transiting planet as TOI-5624 f and determined its minimum mass to be Mf sin if = 13.0 ± 3.7 M⊕. Conclusions. 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 (>3σ) of the mass for at least three planets has previously been 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.