I have documented a workaround for the said issue, which is applicable to ASUS notebook PC that I have been using since late-2014.
The workaround based on what I have documented in mid-2016:
When the battery is empty, or when notebook PC is unable to power on, connect the notebook PC to the charger and start charging.
Do not boot to Linux. The notebook PC should remain off whilst charging. Wait for 10~30 minutes, depending on charging rate.
Afterwards, turn on the notebook PC and boot to Linux. Login to the desktop and confirm that the battery is charging from the Power Manager or using
upower -d command in Terminal.
The important point is notebook PC should be turned off when charging battery from empty, otherwise charging will not start. Even suspend will prevent the battery from charging at all. I have observed that again and again, whenever I had left my notebook PC unused for weeks.
Note: Documented for ASUS X200MA, running Xubuntu 16.04 LTS 64-bit.
Despite the workaround has worked for my notebook PC, I have thought of other issues that may be related to the question:
The latter may be influenced by proprietary firmware for power management, which can only be fully utilized by Windows operating system but not Linux. This may explain the condition when battery is empty, charging may start when running Windows but does not start when running Linux.
There is one critical advice: Lithium ion battery should not be fully discharged or should not be used until empty (see Battery treatment on ThinkWiki).
Regardless of running Windows or Linux, the operating system supposedly has sane default settings to prevent the battery from fully discharged. As a result, the system will automatically suspend, hibernate or even shut down when battery level has reached to certain percentage. This however, may be changed by users.
TL;DR Forget Windows and Linux. Charge the battery from empty when the power is off. Battery should not be fully discharged because of potential damage or reduced battery life.