Recent advancements of the experiment to search for $2K$-capture in $^{124}$Xe using a Large Low-background Proportional Counter
A new method for correcting the charge signal from the large proportional counter is proposed. The correction method makes it possible to take into account the loss of primary ionization electrons in the presence of weakly increasing micro impurities of electronegative gases during long-term measurements. It allows increasing the efficiency of the selection of useful events and thereby increasing the sensitivity to $2\nu2K$-decay in $^{124}$Xe at the level of $2\times10^{22}$ from over six years of measurements using a large proportional counter. Analysis of the calibration data, taking into account the new method for correcting the energy release in the detector, made it possible to register events from double $K$ photoionization by a single photon.