In this article, I will present the status of our calculation of the difference between the masses of the long- and short-lived neutral K mesons, $\Delta m_K$ predicted by the Standard Model. This calculation is performed on an ensemble of 152, $64^3 \times 128$ gauge configurations with an inverse lattice spacing of 2.36 GeV and physical quark masses. The results from different methods of analysis and our progress toward obtaining a final result will be discussed.