  The large-$N$ limit of ABJM theory is holographically dual to M-theory on
AdS$_4\times S^7/\mathbb{Z}_k$. The 3-sphere partition function has been
obtained via localization, and its leading behavior $F_{\text{ABJM}}^{(0)}\sim
k^{1/2}N^{3/2}$ is exactly reproduced in the dual theory by tree-level
supergravity. We extend this comparison to the sub-leading $\mathcal O(N^0)$
order by computing the one-loop supergravity free energy as a function of $k$
and comparing it with the ABJM result. Curiously, we find that the expressions
do not match, with $F_{\text{SUGRA}}^{(1)}\sim k^6$, while
$F_{\text{ABJM}}^{(1)}\sim k^2$. This suggests that the low-energy
approximation $Z_{\text{M-theory}}=Z_{\text{SUGRA}}$ breaks down at one-loop
order.
