Connection of two open drain outputs to control common LEDs by PIC32MZ
No it will not work because the LEDs will hold the input voltage above 2V. Also note that you've chosen to run the LEDs at a higher current than the battery charger chip guarantees the output voltage at. Even at 4mA only 1V 'low' is guaranteed.
The PIC32MZ requires input voltage for a '0' of less than 0.2Vdd and for a '1' more than 0.8Vdd. You also should provide a bit of margin.
Easiest thing would be to pull the pins up to Vdd with something like 10K, and drive the LEDs with a few MOSFETs or a CMOS inverter or gate package.