  The observed mass excesses of analog nuclear states with the same mass number
$A$ and isospin $T$ can be used to test the isobaric multiplet mass equation
(IMME), which has, in most cases, been validated to a high degree of precision.
A recent measurement [Kankainen et al., Phys. Rev. C 93 041304(R) (2016)] of
the ground-state mass of $^{31}$Cl led to a substantial breakdown of the IMME
for the lowest $A = 31, T = 3/2$ quartet. The second-lowest $A = 31, T = 3/2$
quartet is not complete, due to uncertainties associated with the identity of
the $^{31}$S member state. Using a fast $^{31}$Cl beam implanted into a plastic
scintillator and a high-purity Ge $\gamma$-ray detection array, $\gamma$ rays
from the $^{31}$Cl$(\beta\gamma)$$^{31}$S sequence were measured. Shell-model
calculations using USDB and the recently-developed USDE interactions were
performed for comparison. Isospin mixing between the $^{31}$S isobaric analog
state (IAS) at 6279.0(6) keV and a nearby state at 6390.2(7) keV was observed.
The second $T = 3/2$ state in $^{31}$S was observed at $E_x = 7050.0(8)$ keV.
Isospin mixing in $^{31}$S does not by itself explain the IMME breakdown in the
lowest quartet, but it likely points to similar isospin mixing in the mirror
nucleus $^{31}$P, which would result in a perturbation of the $^{31}$P IAS
energy. USDB and USDE calculations both predict candidate $^{31}$P states
responsible for the mixing in the energy region slightly above $E_x = 6400$
keV. The second quartet has been completed thanks to the identification of the
second $^{31}$S $T = 3/2$ state, and the IMME is validated in this quartet.
