The MAX31341 has a power management mode that monitors the supply voltage on VCC and backup battery voltage connected to AIN and determines which source is used as the internal power supply (please refer to the datasheet for detailed operation). In power management auto mode, pin AIN is connected to the backup battery, D_MODE[1:0]=01, D_MAN_SEL=0.
When the main power VCC is lost, the internal comparator still needs some time to make the comparison of VCC vs VTH and VCC vs AIN. The maximum falling slew rate of the power supply is 1.4V/msec specified in the datasheet. So if the power supply falls faster than this, the comparator will not have time to change from VCC to AIN for the source. The part will be turned off temporarily and the data will not be retained.
In order to deal with fast falling slew rate , adding a capacitor at VCC might solve the problem. The capacitor value should be selected to make sure the falling slew rate is slower than 1.4V/ms mentioned above.