The Central Molecular Zone (CMZ), located in the centre of the Milky Way, is a roughly cylindrical structure of molecular gas extending up to parsecs around the supermassive black hole Sagittarius A$^{*}$. The average H$_2$ ionisation rate in the CMZ is estimated to be $2\times 10^{-14}~\rm{s}^{-1}$, which is 2–3 orders of magnitude higher than anywhere else in the Galaxy. Due to the high gas density in this region, electromagnetic radiation is rapidly absorbed, leaving low-energy cosmic rays (CRs) as the only effective ionising agents. Hence, a high CR density has been invoked to explain such high ionisation rates. However, a corresponding excess in $\gamma$-rays, which would result from interactions of high-energy CRs, has not been observed. This suggests that the supposed excess exists only in the low-energy CR spectrum. To constrain this unknown low-energy component, we first derive the high-energy CR injection spectra using $\gamma$-ray and radio data, to which we add various low-energy components. We then propagate these injection spectra by numerically solving the CR transport equation using a Crank-Nicolson scheme. Testing multiple CR injection scenarios, we find that the energy required to sustain the observed ionisation rates is excessively high in every case. We conclude that CRs cannot be the exclusive ionising agents in the CMZ.