  If the mass excess of neutron-deficient nuclei and their neutron-rich mirror
partners are both known, it can be shown that deviations of the Isobaric Mass
Multiplet Equation (IMME) in the form of a cubic term can be probed. Such a
cubic term was probed by using the atomic mass of neutron-rich magnesium
isotopes measured using the TITAN Penning trap and the recently measured
proton-separation energies of $^{29}$Cl and $^{30}$Ar. The atomic mass of
$^{27}$Mg was found to be within 1.6$\sigma$ of the value stated in the Atomic
Mass Evaluation. The atomic masses of $^{28,29}$Mg were measured to be both
within 1$\sigma$, while being 8 and 34 times more precise, respectively. Using
the $^{29}$Mg mass excess and previous measurements of $^{29}$Cl we uncovered a
cubic coefficient of $d$ = 28(7) keV, which is the largest known cubic
coefficient of the IMME. This departure, however, could also be caused by
experimental data with unknown systematic errors. Hence there is a need to
confirm the mass excess of $^{28}$S and the one-neutron separation energy of
$^{29}$Cl, which have both come from a single measurement. Finally, our results
were compared to ab initio calculations from the valence-space in-medium
similarity renormalization group, resulting in a good agreement.
