If you look at the sch that you will see the charger IC is be short circuit by a diode, so now when you connect it to USB(shut off) then the battery will be connect to the USB VBUS~ then the 5V-0.7V=4.3 V voltage direct charge the battery~ it’s not a good idea to do so, because this way cannot charge the battery to wholely full , but if we move away the diode , that the charger IC work~ you can charger the battery to full , but what you lost: you can not just use the USB power without battery~!
now i have used one battery to test ~ i have been using it half a day and now its voltage goes down to 2.9V so the DSO Nano considers that it’s low power and cannot be turned on now. and i will charger it for 6 houres that you try to test how many energy can be charger into it , but now just 1 hour it seen the voltage go back to 3.95V ~ and jsut see can it full up to more than 4V , and i will test how long can it work again … if it work naturally , that may be the battery that you use is bad … 