ADP5065 datasheet advertises "Operating input voltage from 4.0 V up to 5.5 V " but it only starts charging the battery after VIN > 4.85V. Even when we have set:
Fast charge current programming bus 550mA (the minimum)
Termination voltage programming bus 4.2V
VINx pin input current-limit programming bus 500 mA
System Consumption is only about 60mA