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Isobaric quintets provide the best test of the isobaric multiplet mass equation (IMME) and can uniquely identify higher order corrections suggestive of isospin symmetry breaking effects in the nuclear Hamiltonian. The Generalized IMME (GIMME) is a novel microscopic interaction theory that predicts an extension to the quadratic form of the IMME. Only the $A=20, 32$ $T=2$ quintets have the exotic $T_z = -2$ member ground state mass determined to high-precision by Penning trap mass spectrometry. In this work, we establish $A=36$ as the third high-precision $T=2$ isobaric quintet with the $T_z = -2$ member ground state mass measured by Penning trap mass spectrometry and provide the first test of the predictive power of the GIMME. A radioactive beam of neutron-deficient $^{36}$Ca was produced by projectile fragmentation at the National Superconducting Cyclotron Laboratory. The beam was thermalized and the mass of $^{36}$Ca$^+$ and $^{36}$Ca$^{2+}$ measured by the Time of Flight - Ion Cyclotron Resonance method in the LEBIT 9.4 T Penning trap. We measure the mass excess of $^{36}$Ca to be ME$ = -6483.6(56)$ keV, an improvement in precision by a factor of 6 over the literature value. The new datum is considered together with evaluated nuclear data on the $A=36$, $T=2$ quintet. We find agreement with the quadratic form of the IMME given by isospin symmetry, but only coarse qualitative agreement with predictions of the GIMME. A total of three isobaric quintets have their most exotic members measured by Penning trap mass spectrometry. The GIMME predictions in the $T = 2$ quintet appear to break down for $A = 32$ and greater.