By means of diffuse and inelastic x-ray scattering (DS,IXS), we probe directly the charge-ordering (CO) dynamics in the Verwey system (NaMn$_3$)Mn$_4$O$_{12}$, where a peculiar quadruple perovskite structure with no oxygen disorder stabilizes a nearly full Mn$^{3+}$/Mn$^{4+}$ static charge order at $T_{\rm CO}$=175 K concomitant to a commensurate structural modulation with propagation vector ${\bf q}_{\rm CO}=(\frac{1}{2},\frac{1}{2},0)$. At $T_{\rm CO}$, the IXS spectra unveil a softening of a 35.3 meV phonon at ${\bf q}_{\rm CO}$. Lattice dynamical calculations enable us to attribute this soft phonon to a A$_g$ mode whose polarization matches the Jahn-Teller-like distortion pattern of the structural modulation. This result demonstrates that the Jahn-Teller instability is the driving force of the CO Verwey transition in (NaMn$_3$)Mn$_4$O$_{12}$, thus elucidating a long-standing controversy regarding the mechanism of this transition observed in other mixed-valence systems like magnetite.
