The MAX20057 boost controller cannot start operation with input voltage less than 4.5V(typ). The initial input voltage should be at least 4.5V. However, once the converter starts it continue to work even when the input voltage dips below 4.5V. The converter regulates the output voltage upto a minimum input voltage of 2.05V.
In short, the boost controller can operate with input voltage down to 2.05V once the initial start up is completed.